New Forms of Ergodicity Breaking inQuantum Many-Body Systems

Sanjay Moudgalya

a dissertationpresented to the facultyof princeton university

in candidacy for the degreeof Doctor of Philosophy

recommended for acceptanceby the Department of

Physics

Adviser: B. Andrei Bernevig

September 2020

© Copyright by Sanjay Moudgalya, 2020. All rights reserved.

Abstract

Generic isolated interacting quantum systems are believed to be ergodic, i.e. anysimple initial state evolves to a thermal state at late-times, forming a barrier to theprotection of quantum information. A sufficient condition for the thermalizationof initial states in an isolated quantum system is the Eigenstate ThermalizationHypothesis (ETH). A complete breakdown of ETH is well-known in two kindsof systems: Integrable and Many-Body Localized, where none of the eigenstatessatisfy it. In this dissertation, we introduce two new mechanisms of ergodicitybreaking: Quantum Scars, and Hilbert Space Fragmentation. In both these cases,ETH-violating eigenstates coexist with ETH-satisfying ones, and thus the fate ofan initial state under time-evolution depends on the properties of eigenstates ithas weights on.

We obtain the first analytical examples of quantum scars by solving for severalexcited states in a family of non-integrable quantum systems in one dimension:the AKLT models. These exact eigenstates include an infinite tower of statesfrom the ground state to the highest excited state. The states in the middle of thespectrum obey a logarithmic scaling of entanglement entropy with system size,contrary to the volume-law scaling predicted by ETH. We further show the closeconnections between such quantum scarred models and Parent Hamiltonians ofMatrix Product States, as well as connections to the phenomenon of eta-pairingknown in the context of superconductivity in Hubbard models.

Hilbert space fragmentation occurs in constrained systems with a center-of-massor dipole moment conservation law, which naturally arises within a Landau levelin quantum Hall systems or in systems subjected to a large electric field limit. Weshow that the Hilbert space of such systems fractures into several dynamicallydisconnected Krylov subspaces that are not distinguished by simple symmetries,and thus constitute a violation of conventional ETH. We show that ETH can bemodified to apply to each connected subspace separately, and that large integrableand non-integrable subspaces can co-exist within the same system.

iii

Contents

Abstract iii

0 Prelimanaries 10.1 Quantum Thermalization . . . . . . . . . . . . . . . . . . . . . . . 60.2 Matrix Product States and Entanglement . . . . . . . . . . . . . . 120.3 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 18

1 Excited States of the AKLT Model 211.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.2 The Spin-1 AKLT Model . . . . . . . . . . . . . . . . . . . . . . . 231.3 Exact States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.4 Exact Low energy Excited states of the Spin-1 AKLT Model . . . 291.5 Mid-Spectrum Exact States . . . . . . . . . . . . . . . . . . . . . 351.6 1D Spin-S AKLT Models with S > 1 . . . . . . . . . . . . . . . . 431.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2 Entanglement of Quasiparticle Excited States 562.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.2 MPS and MPO in the AKLT models . . . . . . . . . . . . . . . . 592.3 MPO × MPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.4 Single-mode Excitations . . . . . . . . . . . . . . . . . . . . . . . 742.5 Beyond Single-Mode Excitations . . . . . . . . . . . . . . . . . . . 812.6 Tower of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.7 Implications for ETH . . . . . . . . . . . . . . . . . . . . . . . . . 922.8 Entanglement Spectra Degenarcies and Finite-Size Effects . . . . 942.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3 Quantum Scars from Matrix Product States 1073.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.2 MPS description of Quasiparticles . . . . . . . . . . . . . . . . . . 1093.3 Parent Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 1133.4 Quantum Scarred Hamiltonians . . . . . . . . . . . . . . . . . . . 117

iv

3.5 New Families of AKLT-like Quantum Scarred Hamiltonians . . . . 1253.6 New Type of Quantum Scars: Two-Site Quasiparticle Operators . 1293.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

4 Quantum Scars from Spectrum Generating Algebras 1414.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.2 Review of η-pairing in the Hubbard Model . . . . . . . . . . . . . 1444.3 η-pairing on arbitrary graphs . . . . . . . . . . . . . . . . . . . . 1464.4 Examples of η-pairing . . . . . . . . . . . . . . . . . . . . . . . . 1494.5 Quantum Many-Body Scars from the Hubbard Model . . . . . . . 1524.6 Connections to Quantum Scars . . . . . . . . . . . . . . . . . . . 1574.7 D dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594.8 RSGA and Quantum Scarred Models . . . . . . . . . . . . . . . . 1624.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5 Hilbert Space Fragmentation 1685.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1685.2 Model and its symmetries . . . . . . . . . . . . . . . . . . . . . . 1705.3 Hamiltonian at 1/2 filling . . . . . . . . . . . . . . . . . . . . . . 1745.4 Krylov Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1785.5 Integrable subspaces . . . . . . . . . . . . . . . . . . . . . . . . . 1845.6 Non-integrable subspaces and Krylov-Restricted ETH . . . . . . . 1965.7 Quasilocalization from Thermalization . . . . . . . . . . . . . . . 2005.8 Conclusions and Open Questions . . . . . . . . . . . . . . . . . . 205

6 Approximate Quantum Scars in a Fractured Model 2086.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2086.2 Effective Spin-Chains and Constrained Hilbert spaces . . . . . . . 2106.3 Symmetries and Non-integrability of the Effective Hamiltonians . 2246.4 Quantum Many-Body Scars . . . . . . . . . . . . . . . . . . . . . 2266.5 Forward Scattering Approximation . . . . . . . . . . . . . . . . . 2286.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

Appendix A Algebra of dimers 238A.1 Commutation relations . . . . . . . . . . . . . . . . . . . . . . . . 238A.2 Dimer basis states and scattering rules for the spin-1 AKLT model 240A.3 Dimer basis scattering rules for spin-S AKLT basis states . . . . . 243

Appendix B Review of Matrix Product Operators 248B.1 Matrix Product Operators . . . . . . . . . . . . . . . . . . . . . . 248B.2 Jordan normal form of block upper triangular matrices . . . . . . 253

v

Appendix C Embedding Quasiparticles using MPS Subspaces 259C.1 Total Angular Momentum Eigenstates . . . . . . . . . . . . . . . 260C.2 Examples of A and B subspaces for the AKLT-like MPS . . . . . 260C.3 Single-Site Quasiparticle Exact Eigenstates in the MPS Language 262C.4 SU(2) Multiplet of the Spin-2 Magnon for the AKLT chain . . . . 265C.5 Examples of A and B subspaces for the Potts-like MPS . . . . . . 267

Appendix D Eta pairing and RSGAs 269D.1 Useful Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269D.2 η-pairing with disorder and spin-orbit coupling . . . . . . . . . . . 270D.3 Tower of States from (Restricted) Spectrum Generating Algebras 271

Appendix E Physical Origins of the Pair-Hopping Hamiltonian 275E.1 Bloch Many-Body Localization . . . . . . . . . . . . . . . . . . . 276E.2 Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . 287

References 315

vi

Listing of figures

1.1 Level Statistics of the spin-1 AKLT Model . . . . . . . . . . . . . 251.2 AKLT Ground state. The big circles are physical spin-1 s and the

smaller circles within the spin-1 s are spin-1/2 Schwinger bosons.Symmetric combinations of the Schwinger bosons on each site formthe physical spin-1. The lines joining the Schwinger bosons repre-sent singlets. |G〉 with periodic boundary conditions . . . . . . . . 25

1.3 Arovas A State Dimer Configuration . . . . . . . . . . . . . . . . 321.4 Arovas B State Dimer Configurations . . . . . . . . . . . . . . . . 331.5 Spin-2 Magnon Dimer Configuration . . . . . . . . . . . . . . . . 351.6 Tower of States Dimer Configuration . . . . . . . . . . . . . . . . 371.7 Destructive Interference of Spin-2 Magnons . . . . . . . . . . . . . 391.8 Position of the AKLT tower of states in the spectrum . . . . . . . 411.9 Spin-2 AKLT Ground State Dimer Configuration . . . . . . . . . 451.10 Spin-2 AKLT Arovas B State Dimer Configurations . . . . . . . . 471.11 Spin-4 Magnon of Spin-2 AKLT Model . . . . . . . . . . . . . . . 511.12 Scattering Rules of the Spin-1 AKLT Model . . . . . . . . . . . . 55

2.1 Spin-1 AKLT Ground State . . . . . . . . . . . . . . . . . . . . . 592.2 Spin-2 AKLT Ground State . . . . . . . . . . . . . . . . . . . . . 612.3 Entanglement entropy of AKLT eigenstates in the quantum number

sectors with scar states . . . . . . . . . . . . . . . . . . . . . . . . 922.4 Entanglement Spectra of AKLT Towers of States . . . . . . . . . 102

5.1 Evidence of Krylov-Restricted ETH . . . . . . . . . . . . . . . . . 1975.2 Quasilocalization from Thermalization . . . . . . . . . . . . . . . 201

6.1 Level Statistics of the ν = 2/5 Krylov subspace of the pair-hoppingHamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

6.2 Hilbert Space Graph for the ν = 2/5 Krylov subspace of pair-hopping Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 224

6.3 Anomalous Dynamics of the Pair-Hopping Hamiltonian with ν = 2/52296.4 Approximate Scars of the Pair-Hopping Hamiltonian with ν = 2/5 231

vii

6.5 Stability of the ν = 2/5 scars . . . . . . . . . . . . . . . . . . . . 233

A.1 Two types of singlet configurations around a bond i, j. . . . . . 241A.2 Types of non-singlet configurations around a bond i, j . . . . . 242A.3 Scattering of S = 2 dimer configurations . . . . . . . . . . . . . . 244

E.1 Bloch Many-Body Localization . . . . . . . . . . . . . . . . . . . 285

viii

To my family,

ix

Acknowledgments

This thesis would have not been possible without the enormous support of a lotof people, and I am woefully short of words to express my gratitude.

Firstly, I would like to thank my advisor Andrei Bernevig for his guidance,encouragement, and support. Andrei’s vision and strong intuition shaped thisthesis. His infectious enthusiasm and optimism taught me that physics could bea lot of fun, and his high standards of rigor constantly motivated me and greatlyenhanced the clarity of my thoughts and scientific writing. I am also indebted tohim for generously giving me the freedom and funding to pursue my own ideas andcollaborations, due to which my grad school experience turned out to be exactlyas I wanted it to be.

Next, I would like to thank Nicolas Regnault, who played a role akin to myprimary advisor’s. Nicolas’ humor and encouragement has made working on hardproblems exciting. His patient comments have greatly improved my writing, pre-sentation, and computational skills. I am also extremely grateful for his invitationto spend a summer in ENS Paris, which has been one of the most fun and pro-ductive times during graduate school.

I am also deeply indebted to Shivaji Sondhi for his guidance and for the manycollaborations. Working closely with him on a wide variety of problems has beenan incredible learning experience and helped me broaden my view of physics.In addition, his constant encouragement and close supervision right from thebeginning has made navigating my way through grad school easier.

I have further benefited a lot from several others in the physics community. Iwould particularly like to express my immense gratitude to Frank Pollmann forintroducing me to research in condensed matter physics, and for his extraordinarysupport during graduate school applications without which this thesis would notexist. I am greatly indebted to David Huse for several discussions over the yearson several problems, and I thank him for sharing his intuition and providing mewith a glimpse of the vast ocean of his knowledge. In addition, I thank DanielArovas, Amos Chan, Trithep Devakul, Akash Goel, Huan He, Curt von Keyser-lingk, Alan Morningstar, Mike Zaletel, Yunqin Zheng, and particularly VedikaKhemani and Abhinav Prem for several engaging discussions and collaborationsthat have greatly sharpened my thinking. I also acknowledge Edward O’Brien,

x

Paul Fendley, Rahul Nandkishore, Abhinav Prem, and Stephan Rachel for theircontributions to the work presented in this thesis. I am also grateful to Igor Kle-banov and Ali Yazdani for agreeing to serve on my thesis committee, and to BobAustin for supervising my experimental project. I further thank Kate Brosowskyand Toni Sarchi for smoothly taking care of my administrative needs, and themembers of the Department of Physics for creating a wonderfully conducive en-vironment for research.

These past five years would have been devoid of fun if not for my friends andfamily. I am particularly indebted to Sravya whose amazing company single-handedly lit up everyday life in Princeton. I am also grateful to all my otherfriends: Ajesh, Akash, Akash, Akshay, Akshay, Arjun, Deeksha, Digvijay, Di-vya, Gopi, Karan, Nikunj, Niranjani, Pranav, Prashanth, Sumegha, Vivek amongothers, for the regular trips, sports, movies, lunches and dinners, that broke themonotony of life. I thank Nani Uncle and Suji Aunty for helping me settle downin Princeton, and for regularly providing home-cooked food. Last, but not theleast, endless thanks go to my incredible family, to whom this thesis is dedicated.I would like to acknowledge my maternal grandmother for all the delicious foodand the boundless love, and my uncles for their guidance. The extent of gratitudeI owe my parents and sister is such that I cannot even begin to properly thankthem. Their never-wavering affection, support, and faith in me shaped me intowho I am today. I would especially like to thank my sister Sanjana for the fre-quent calls that allowed me to briefly detach myself from work and fully unwind.

xi

0Prelimanaries

Rapid advances in the coherent control and manipulation of cold atoms have

enabled experiments to study the non-equilibrium dynamics of closed quantum

many-body systems [1–7]. Consequently, the question of how (and whether) an

arbitrary quantum state achieves thermal equilibrium while evolving under uni-

tary dynamics has moved to the forefront of contemporary research. Besides,

the discovery of the Sachdev-Ye-Kitaev (SYK) model and its connection to black

holes has garnered the interest of multiple physics communities in the dynamics of

1

simple quantum mechanical systems [8–13]. The dynamics of a quantum system

is tied to the properties of all its energy eigenstates and not only to the ground

state features, the latter being the primary focus of a large part of condensed

matter physics. It is thus important to understand the structure of highly excited

states of quantum systems.

An important theoretical development along these lines is the Eigenstate Ther-

malization Hypothesis (ETH) [14–17], which states that, as far as expectation

values of local observables are concerned, eigenstates of an ergodic system display

thermal behaviour [18–20]. However, in spite of a vast amount of literature, ETH

remains unproven even after almost three decades since its introduction and is only

supported by numerical evidence that relies on large scale computation [16–18,21],

which limits the systems we can access and hinders our understanding. A major

obstacle is the lack of exactly solvable examples of highly excited states in the bulk

of the energy spectrum. There do exist several “integrable” models with solvable

highly excited states: free systems, and models solvable via the so-called Bethe

Ansatz [22], the latter including several well-known models in one dimension in-

cluding the XXZ and Hubbard models. However, these integrable models have an

extensive number of conserved quantities and are unfortunately also well known

examples where ETH breaks down [23–25]. For generic non-integrable systems

that are not known to be solvable, typically none of the excited eigenstates can be

obtained analytically, and it is not clear if ETH-satisfying closed-form eigenstates

of a non-integrable, although there is no known physical principles forbidding their

existence. Thus looking for some simple non-integrable models where a partial or

complete analytical description beyond the low energy states is available would be

a perfect avenue to understand ETH, and this the is primary aim of this thesis.

2

Since thermalization is believed to occur in generic isolated quantum systems,

the breakdown of thermalization is an extremely interesting phenomenon. For

example, the discovery of Many-Body Localization (MBL) [26–29] has opened up

exciting possibilities of realizing exotic equilibrium phases in the non-equilibrium

setting [30–35], and led to the discovery new phases of matter forbidden in equi-

librium such as time crystals [36]. MBL systems appear with the addition of

disorder or quasiperiodic potentials [37], and evade ergodicity even at high en-

ergy densities, retain a memory of their initial conditions in local observables for

arbitrarily long times that leads to rich new physics which has been extensively

studied numerically [21, 38–42]. The complete breakdown of ETH is thus widely

known in two kinds of systems: integrable, where all highly excited eigenstates

have a quasiparticle description, and many-body localized where all highly excited

eigenstates are localized. An important open question is whether similar phenom-

ena, e.g. violation of ETH or memory of initial conditions at long times, can occur

in translation invariant non-integrable systems [43–52]. In this thesis, we describe

a new type of ETH violation that has recently been added to this list: Quantum

Many-Body Scars. Unlike other two mechanisms of ETH violation where the en-

tire spectrum of a Hamiltonian violates ETH, quantum scarred models consist of

some ETH-violating eigenstates in an otherwise ETH-satisfying spectrum. The

first exact examples of quantum scars include a systematic embedding of non-

thermal eigenstates in a thermal spectrum [53], and an equally spaced tower of

ETH-violating eigenstates discovered in the spin-1 Affleck-Kennedy-Lieb-Tasaki

(AKLT) chain [54–56], which we discuss in Chapters 1 and 2. Similar towers

of eigenstates were subsequently discovered in several families of models [57–60].

Recent works [61–65] have revealed rich algebraic structures in models exhibiting

3

towers of quantum scars, and have established connections to Matrix Product

States and Spectrum Generating Algebras, which are discussed in Chapters 4 and

5 respectively. Concurrent to these developments, an experiment on a 1D chain of

Rydberg atoms observed persistent revivals upon quenching the system from cer-

tain initial conditions, while other initial conditions led to the system thermalizing

rapidly [6]. This striking dependence on initial conditions was numerically demon-

strated to be caused by a vanishing number of non-thermal states that co-exist

with an otherwise thermal spectrum [66–70], dubbed “quantum many-body scars”.

Such anomalous dynamics were traced numerically to the initial state having a

high overlap with an equally spaced tower of apparently ETH-violating eigen-

states in the so-called PXP model [66, 71, 72], a Rydberg-blockade Hamiltonian

modelling this experiment [66, 68]. Various attempts to explain the anomalous

dynamics phenomenon include connections to classical scars on an emergent clas-

sical manifold [70, 73], proximity to integrability [74], existence of momentum-π

quasiparticles on top of an exact [75] or approximate [76] eigenstate, and construc-

tion of parent Hamiltonians with almost-perfect revivals [69], including by using

ideas from Lie algebras [62]. Furthermore, the presence of approximate revivals

has also been demonstrated numerically in a variety of other models resembling

the PXP model [70, 77–79]. In addition, there are several theoretical and numer-

ical works explore systems showing anomalous dynamics and the phenomenology

of quantum scars [60, 72, 80–95]. These discoveries thus reveal new possibilities

for quantum dynamics which may occur between the extremes of thermalization

and the complete breaking of ergodicity. They constitute a violation of “strong

ETH” [96, 97], which states that all eigenstates in the middle of the spectrum

obey ETH but they still satisfy “weak ETH”, where almost all eigenstates obey

4

ETH.

In addition to quantum scars, another mechanism of ergodicity breaking was

recently discovered in constrained quantum systems, dubbed as Hilbert space

fragmentation [86], Hilbert space shattering [87], or Krylov fracture [78, 88]. In

particular, quantum systems with a dipole moment in addition to charge con-

serving quantum systems exhibit a strong dependence of initial product state,

first noted in studies of operator spreading in random circuits with a dipole mo-

ment symmetry [98]. Such a phenomenon can be traced back to the existence

of exponentially many “Krylov subspaces”, i.e. disconnected components of the

Hilbert space such that product states in one component cannot evolve into any

other component. While such phenomenology always occurs if the said product

state is invariant under a symmetry of the Hamiltonian, these disconnected com-

ponents exist in fractured systems even within a given symmetry sector. While

some of these fractured systems violate both strong and weak ETH with respect

to the full Hilbert space, they appear to satisfy weak ETH when viewed within

each Krylov subspace of the Hamiltonian, similar to a phenomenon known as re-

stricted thermalization in classical kinetically constrained models [99]. We discuss

these aspects in Chapters 5 and 6, along with their close connections to several

physical systems.

The rest of this chapter is organized as follows. In Sec. 0.1, we review the

Eigenstate Thermalization Hypothesis and its formal statement, and review mech-

anism of its breakdown. In Sec. 0.2, we review concept of entanglement in quan-

tum many-body systems, and discuss its connections to so-called Matrix Product

States (MPS), which plays an important role in the study of exactly solved ex-

amples of quantum scars. Finally in Sec. 0.3, we provide an brief outline of the

5

remaining chapters in this thesis.

0.1 Quantum Thermalization

We now precisely define what it means for an isolated quantum system to be

thermal. For the sake of concreteness, consider a system with L spins, with the

wavefunction |ψ(0)〉 at time zero. In typical experimental settings, |ψ(0)〉 is a

ground state of a known Hamiltonian or a product state. An isolated quantum

system evolves the state unitarily, i.e. if the Hamiltonian is H, the state of the

full system at time t reads

|ψ(t)〉 = e−iHt |ψ(0)〉 . (1)

Considering a bipartition of the system into subsystems A and B, with LA and

LB spins each such that LA LB. We can define a reduced density matrix of the

full state |ψ〉 over the subsystem A as

ρA ≡ TrB (|ψ〉 〈ψ|) , (2)

An isolated quantum system without any other symmetries is said to be thermal

if the reduced density matrix of any small subsystem A evolves to a Gibbs density

matrix:

limt→∞

ρA(t) = TrB (ρeq) ≈ ρeqA , ρeq =1

Ze−βH (3)

where H is the Hamiltonian, Z is a partition function for the subsystem, and β

is an “inverse-temperature” associated with the state, which we discuss below.

6

In particular, Eq. (3) implies that the rest of the subsystem B acts as a thermal

bath for the small system A. In the presence of additional symmetries, Eq. (3) is

modified to include a grand canonical ensemble formed by the symmetries. For

example, with particle number conservation, we expect

limt→∞

ρA(t) = TrB (ρeq) , ρeq =1

Ze−βH−µN , (4)

where µ is a chemical potential and N is the particle number operator. These

conditions on the dynamics of states have a direct implication on the structure of

eigenstates of the system, which we now discuss.

0.1.1 Eigenstate Thermalization Hypothesis (ETH)

If any initial state of a system thermalizes, the eigenstates of a system should

also thermalize, and we thus arrive at the Eigenstate Thermalization Hypothesis

(ETH), which loosely states that any eigenstate of the Hamiltonian is thermal.

That is, the reduced density matrix of an eigenstate with energy Eα over a small

subsystem A should also be the Gibbs density matrix over the subsystem. As

a consequence, expectation values of (sums of) few-body operators O within an

eigenstate should match their thermal expectation values:

〈Eα| O |Eα〉 =1

ZTr(Oe−βαH

), (5)

7

where βα is the inverse temperature associated with the eigenstate |Eα〉, which

can be determined using Eq. (5) with O = H:

Eα =

∑γ

Eγe−βαEγ

∑γ

e−βαEγ

. (6)

Note that while βα is determined using Eq. (6), ETH states that Eq. (5) holds for

any local operator O with the same value of βα.

ETH is also motivated by considering a thermodynamic point of view for the

expectation values of operators. An initial state can be expanded in the energy

eigenbasis of H as

|ψ(0)〉 =∑α

cα |Eα〉. (7)

We consider an initial state has with typical energy E and small energy variance

∆,

〈ψ(0)|H |ψ(0)〉 = E,√〈ψ(0)|H2 |ψ(0)〉 − E2 = ∆ W, (8)

where W is the bandwidth of the system, which is typically the case for product

initial states and local Hamiltonians [16]. The expectation value of a local operator

O as a function of time thus reads

O(t) ≡ 〈ψ(t)| O |ψ(t)〉 =∑α

|cα|2Oαα +∑α =β

c∗αcβOαβei(Eα−Eβ)t, (9)

where Oαβ = 〈Eα| O |Eβ〉. Due to phase cancellation of the off-diagonal terms in

Eq. (9), the long-time average is determined only by the average in the “diagonal

8

ensemble”:

limT→∞

1

T

∫ T

0

dt O(t) =∑α

|cα|2Oαα. (10)

In a thermalizing system, we expect the long-time average to be equal to the

expectation value of a local operator in a microcanonical ensemble around energy

E, requiring ∑Eα∈[E−∆,E+∆]

Oαα =∑α

|cα|2Oαα (11)

This suggests that Oαα on the RHS is only a function of the energy E rather than

the eigenstate Eα. These arguments, along with many other motivations [15], lead

to a formal conjecture on the matrix elements of local operators in the energy

eigenstates of a non-integrable model take the form [18]

〈Em| O |En〉 = O (E) δm,n +Rm,ne−S(E)/2fO (E, ω) , (12)

where O is a local operator that is invariant under the symmetries of the Hamil-

tonian, |Em〉 and |En〉 are the energy eigenstates with energies Em and En with

the same symmetry quantum numbers, E = (Em + En) /2, ω = Em−En, Rm,n is

a random variable with zero mean and unit variance, O (E) is a smooth function

of E and represents the thermal expectation value of O at energy E, fO (E, ω) is

a smooth function of E and ω which do not scale with the system size [18], and

S (E) is the thermodynamic entropy at energy E. Note that the thermal value

is typically determined by computing the microcanonical average, i.e. averaging

the eigenstate expectation values 〈E| O |E〉 over a small energy window ∆ that

corresponds to the Thouless energy scale [100]. In Eq. (12), since S(E) ∼ logD

for states in the middle of the spectrum, where D is the Hilbert space dimension,

9

the standard deviation of expectation values of operators in the eigenstates is ex-

pected to scale as ∼ 1/√D for eigenstates in the middle of the spectrum, which

forms a standard diagnostic of ETH [88,100, 101].

0.1.2 Diagnostics of Thermalization

We now review some standard diagnostics of non-integrability of a quantum

Hamiltonian used in the literature.

Energy level statistics

Non-integrable Hamiltonians generically lack symmetries and thus neighboring

eigenvalues repel each other. This observation leads to a remarkably accurate

prediction for the statistics of nearest-neighboring energy differences in the “un-

folded” spectrum, i.e. the statistics of sn = (En+1 − En)/E, where En’s are the

energy levels and E is the mean energy level spacing in the vicinity of En [102].

It has been numerically verified for several non-integrable models that sn follows

a Wigner-Dyson distribution for a non-integrable system that thermalizes [16,

38, 103] whereas sn in a systems with several symmetries (e.g. integrable/MBL)

system exhibits a Poisson distribution. The appearance of Wigner-Dyson statis-

tics is thus taken to be one of the defining feature of quantum chaos, and it

can be explained semiclassically in certain quantum systems with a well-defined

classical limit. However, the level spacing assumes a flat density of states in

the vicinity of levels being studied and thus requires an “unfolding” of the spec-

trum, which can be tedious to implement reliably. Hence a level spacing ra-

tio rn = min(δn, δn+1)/max(δn, δn+1), where δn = En+1 − En was discussed in

Ref. [104, 105], which does not depend on details of unfolding the spectrum. A

10

simple parameter that detects the difference between the two possible level-spacing

distributions is hence the average level spacing ratio 〈r〉, which shows the values of

〈r〉 ≈ 0.53 for non-integrable systems with time-reversal symmetry, and 〈r〉 ≈ 0.38

for Poisson distributed energy level spacings. Note that for non-integrable Hamil-

tonians with a few additional symmetries (e.g. particle number), level repulsion

is expected to show in the distribution of energy levels within a symmetry sector.

It is thus important to determine all the symmetries of the system.

Entanglement Entropy

Another probe of non-integrability of a system is the entanglement entropy of

excited states of a system [38]. Given a bipartition of a system into regions A and

B, we first define a Schmidt decomposition of a state |ψ〉 as

|ψ〉 =χ∑α=1

λα |ψα〉A |ψα〉B, (13)

where |ψα〉A and |ψα〉B are orthonormal sets of wavefunctions on the subsys-

tems A and B respectively, and χ is known as the Schmidt rank of the wavefunc-

tion. Note that for a normalized state |ψ〉, we always obtainχ∑α=1

λ2α = 1. The

von Neumann entropy S of the state |ψ〉 over this bipartition is defined as the

Shannon entropy of the Schmidt coefficients, i.e.

S ≡ −χ∑α=1

λ2α log λ2α = −Tr ρA log ρA (14)

where ρA is the reduced density matrix over subsystem A (see Eq. (2)). The

entanglement entropy is a quantity that is widely used in several contexts in

11

condensed matter physics [106]. Ground states of gapped quantum many-body

systems are known to exhibit a so-called “area-law” scaling of the entanglement

entropy, where S scales with the area of the subsystem A. In one-dimension,

that scaling implies that S is independent of the subregion size, which has led to

tremendous breakthroughs in the simulation of ground states of one-dimensional

systems via the concept of Matrix Product States (MPS), which we discuss in

the next section. For ground states of critical gapless systems, the entanglement

entropy typically only exhibits a logarithmic violation of the area-law, i.e. S scales

with the area times the logarithm of the volume of the subsystem A.

For highly excited states of non-integrable models, ETH predicts a “volume

law” scaling of S i.e. it scales linearly with the size of the subsystem A. This is a

direct consquence of the reduced density matrix in Eq. (2). In fact, for states in

the middle of the spectrum, we obtain β = 0 in Eq. (2), an thus the EE is equal to

Sth, the average EE of a random state in Hilbert space [107] which is close to the

maximum possible entropy Smax. For a system with L spin-12’s and L/2 spin-1

2’s

in subsystem A, these read

Sth =L log 2− 1

2, Smax =

L log 2

2. (15)

0.2 Matrix Product States and Entanglement

In this section we provide a basic introduction to the Matrix Product States (MPS)

and their properties. We invite readers not familiar with MPS to read numerous

reviews and lecture notes in the literature [108–111]. We consider a spin-S chain

with L sites. A simple many-body basis for the system is made of the product

12

states |m1m2 . . .mL〉 where mi = −S,−S +1, . . . , S − 1, S is the projection along

the z-axis of the spin at site i. Any wavefunction of the many-body Hilbert space

can be decomposed as

|ψ〉 =∑

m1,m2,...,mL

cm1m2...mL|m1m2 . . .mL〉. (16)

In all generality, the coefficients cm1,m2...mLcan always be written as an MPS, [112]

i.e.,

cm1,m2...mL= [blA

TA

[m1]1 A

[m2]2 . . . A

[mL]L brA]. (17)

The state |ψ〉 then reads

|ψ〉 =∑

m1m2...mL

[blATA

[m1]1 . . . A

[mL]L brA] |m1 . . .mL〉. (18)

In Eqs. (17) and (18), A[m1]1 , . . . , A

[mL−1]L−1 and A

[mL]L are χ × χ matrices over an

auxiliary space. χ is the bond-dimension of the MPS and the corresponding

indices are the ancilla. blA and brA are χ-dimensional left and right boundary

vectors that determine the boundary conditions for the wavefunction. The [mi]

are called the physical indices and can take d = 2S + 1 values (d is the physical

dimension, i.e. the dimension of the local physical Hilbert space on site i). In a

compact notation, we can think of the Ai’s as d× χ× χ tensors.

An MPS representation is particularly powerful if the matrices A[mi]i are site-

independent, i.e. A[mi]i = A[mi]. Typically, translation invariant systems admit

such a site independent MPS. Many computations involving an MPS can then be

13

simplified once we introduce the transfer matrix

E =∑m

A[m]∗ ⊗ A[m] (19)

where ∗ denotes complex conjugation and the ⊗ is over the ancilla. The transfer

matrix is thus a χ× χ× χ× χ tensor that can also be viewed as χ2 × χ2 matrix

by grouping the left and right ancilla of the two MPS copies together. The sim-

plification provided by the MPS description can be illustrated by computing the

norm 〈ψ|ψ〉 of the state |ψ〉,

〈ψ|ψ〉 = blT

EELbrE, blE = (blA

∗ ⊗ blA) brE = (brA∗ ⊗ brA). (20)

An MPS representation is said to be in a left (right) canonical form if the largest

left (right) eigenvalue of the transfer matrix E is unique, is equal to 1 (this can

always be obtained by rescaling the B’s) and most importantly the corresponding

left (right) eigenvector is the identity χ×χ matrix. [111] Thus, for a right canonical

MPS, ∑γ,ϵ

Eαβ,γϵδγ,ϵ = δαβ (21)

where δ denotes the Kronecker delta function. However, in general, an MPS

cannot be in both a left and right canonical form simultaneously.

Another useful construction with an MPS is the generalized transfer matrix EO

EO =∑n,m

A[m]∗ ⊗OmnA[n]. (22)

Here O is any single-site operator with matrix elements 〈m| O |n〉 = Omn. EO is

14

useful when computing the expectation value of an operator O acting on a site i,

where

〈ψ| Oi |ψ〉 = blETEi−1EOE

L−ibrE. (23)

Similarly, assuming i < j, the two point function associated with O reads

〈ψ| OiOj |ψ〉 = blETEi−1EOE

j−i−1EOEL−jbrE. (24)

Using Eqs. (23) and (24) for large L, the correlation length ξ of the MPS defined

using

〈OiOj〉 − 〈Oi〉〈Oj〉 ∼ exp

(−|i− j|

ξ

), ξ = − 1

log |ϵ2|, (25)

where ϵ2 is the second largest eigenvalue of the transfer matrix [110]. Note that

−1/ log |ϵ2| is an upper bound for ξ that is saturated unless O has a special struc-

ture. Thus, if the spectrum of the transfer matrix is gapless, the state has an

infinite correlation length. Note that a finite correlation length for an MPS in a

canonical form guarantees that the wavefunction is normalized in the thermody-

namic limit.

0.2.1 Entanglement of MPS

The MPS representation of any wavefunction encodes the entanglement struc-

ture of the wavefunction. Using the MPS representation of |ψ〉 Eq. (18), if the

subsystem A is defined as the set of sites 1, 2, . . . , LA and the region B as

15

LA + 1, LA + 2, . . . , L, the bipartition similar to Eq. (13) can be written as

|ψA〉α =∑

mi,i∈A

[blAT∏l∈A

A[ml]l ]α |mi〉, |ψB〉α =

∑mi,i∈B

[∏l∈B

A[ml]l brA]α |mi〉.

(26)

Note that |ψA〉α and |ψB〉α form complete but not necessarily orthonormal

bases on the subsystems A and B respectively. The reduced density matrix with

respect to such a bipartition is constructed as ρA = TrB |ψ〉 〈ψ|. The eigenvalue

spectrum of − log ρA is the entanglement spectrum and S ≡ −TrA (ρA log ρA) is

the von Neumann entanglement entropy. An alternate way to obtain ρA that is

useful for MPS is through the definition of Gram matrices L and R,

Lαβ = α 〈ψA|ψA〉β = (ET )LAblE, Rαβ = α 〈ψB|ψB〉β = ELBbrE. (27)

In Eq. (27), E is viewed as a χ× χ× χ× χ tensor, blE and brE as χ× χ matrices.

Consequently, L and R are χ× χ matrices. Up to a overall normalization factor,

the reduced density matrix can be expressed in terms of these Gram matrices

as [113], ρA =√LRT

√L, which has the same spectrum as the matrix

ρred = LRT . (28)

Since we are only interested in the spectrum of ρA in this article, we refer to ρred

to be the reduced density matrix of the system even though it is not guaranteed

to be Hermitian. Assuming that the eigenvalue of unit magnitude of the transfer

matrix is non-degenerate, if LA and LB are large, (ET )LA and ELB project onto

eL and eR, the left and right eigenvectors corresponding to the largest eigenvalue

16

of E. Thus,

L = eL(eTLb

lE), R = eR(e

TRb

rE), ρred = eLe

TR. (29)

One should note that the construction of an MPS for a given state is not unique.

Indeed, MPS matrices and boundary vectors redefined as

A[m] = GA[m]G-1 , blM = G-1T blM brM = GbrM (30)

represent the same wavefunction. When constructed in a canonical form, the

bipartition Eq. (26) is the same as a Schmidt decomposition of the state |ψ〉 with

respect to subregions A and B, defined as

|ψ〉 =χs∑α=1

λα |ψsA〉α |ψsB〉α (31)

where |ψsA〉α and |ψsB〉α are sets of orthonormal vectors on the subsystems

A and B respectively and λα are referred to as the Schmidt values and χs is

the number of non-zero Schmidt values (Schmidt rank). The bond dimension

χ of the MPS constructed in the canonical form is the Schmidt rank χs of the

wavefunction |ψ〉. Thus we refer to χs as the optimum bond dimension for an

MPS representation of state |ψ〉. The entanglement entropy then satisfies

S = −χs∑α=1

λ2α log λ2α ≤ logχs (32)

The entanglement entropy of an MPS about a given cut is thus upper-bounded by

logχs. Since the Schmidt decomposition is the optimal bipartition of the system,

χ ≥ χs and hence S ≤ logχ.

17

0.3 Outline of this thesis

The balance of this thesis consists of five chapters and discusses two new ways of

ergodicity breaking in isolated quantum systems: Quantum Scars (Chapters 1-4)

and Hilbert Space Fragmentation (Chapters 5-6).

Chapter 1 is based on Ref. [55], and we propose a novel method to numerically

identify analytically solvable eigenstates within non-integrable models using the

entanglement spectrum of the eigenstates, and use it to obtain exact analytical

expressions for several excited states in the non-integrable integer spin AKLT

models, including an infinite tower of states from the ground state to the high-

est excited state. The eigenstates in these tower could be expressed as multiple

quasiparticles of a fixed momentum dispersing on top of the ground state. Some

states of the tower are in the bulk of the energy spectrum, forming one of the

first examples of solvable eigenstates of a non-integrable model in a region of the

spectrum where ETH is supposed to hold.

Chapter 2 is based on Ref. [56], and we show that the exact eigenstates obtained

in the AKLT models have a simple description using MPS, and we use that de-

scription to obtain analytical results on the entanglement of these eigenstates. In

particular, we show that the states in the middle of the spectrum obey a logarith-

mic scaling of entanglement entropy with system size, contrary to the volume-law

scaling predicted by strong ETH, hence showing a clear violation of strong ETH.

These states thus form the first examples of quantum many-body scars.

Chapter 3 is based on Ref. [63], and we explore the connection between MPS

wavefunctions and quantum scarred Hamiltonians. We provide a method to sys-

tematically search for and construct parent Hamiltonians with towers of exact

18

eigenstates composed of quasiparticles on top of an MPS wavefunction, and we

recover the AKLT chain starting from the MPS of its ground state using this

approach. We derive the most general nearest-neighbor Hamiltonian that shares

the AKLT quasiparticle tower of exact eigenstates, and apply this formalism to

several other simple MPS wavefunctions. As a consequence, we also construct

a scar-preserving deformation that connects the AKLT chain to the integrable

spin-1 pure biquadratic model.

Chapter 4 is based on Ref. [64], where we focus on η-pairing states in Hub-

bard models and explore their connections to quantum many-body scars. Gen-

eralizing the original η-pairing construction to several Hubbard-like models on

arbitrary graphs, we define the concept of a Restricted Spectrum Generating Al-

gebra (RSGA) and give examples of perturbations to the Hubbard-like models

that preserve an equally spaced tower of the original model as eigenstates, whcih

form exact quantum many-body scars. The RSGA framework also explains the

equally spaced towers of eigenstates in several well-known models of quantum

scars, including the AKLT model.

Chapter 5 is based on Ref. [88], and we explore the dynamics of a “pair-hopping

model”, a simple constrained model with a center-of-mass or dipole moment con-

servation law. We find that under time-evolution distinct product states can gen-

erate distinct subspaces that are dynamically disconnected under time-evolution

by the pair-hopping Hamiltonian, a phenomenon that we refer to as Hilbert space

fragmentation or Krylov fracture. These subspaces are interesting due the ab-

sence of any obvious symmetries distinguishing the different subspaces, and they

constitute a violation of conventional ETH. However, we numerically show that

ETH is generically restored when defined with respect to each connected subspace,

19

although we find the coexistence of integrable and non-integrable subspaces.

Chapter 6 is based on Ref. [78], and we focus on certain Krylov subspaces in

the pair-hopping model that are relevant in the quantum Hall effect, and show

the appearance of approximate quantum scars, i.e. a violation of strong ETH

even when defined with respect to each connected subspace. We show this in the

pair-hopping model at one-third filling, where the Hamiltonian restricted to one

of the subspaces exactly maps onto the PXP model, and similar subspaces can be

obtained at other filling factors. These subspaces result in the slow thermalization

of certain charge density wave initial states under the dynamics of the pair-hopping

model.

20

1Excited States of the AKLT Model

1.1 Introduction

For generic non-integrable systems, none of the energy eigenstates can be obtained

analytically. However, the ground state is known exactly for some non-integrable

models with local Hamiltonians, and in this chapter we propose to exploit the

simple structure of such Hamiltonians and look for exactly solvable excited states

in their spectrum. We focus on one such model is the one dimensional spin-1

21

AKLT chain [54, 114], which was first introduced as a simple model to exemplify

the Haldane gap in integer spin chains. Indeed, the ground state of the AKLT

chain can be explicitly built and it belongs to the same universality class as that of

the spin-1 Heisenberg model. Along with its generalizations to higher integer spin

values, it is representative of the Haldane phase [115, 116]. The simplicity of the

ground state of the AKLT model makes it one of the most elegant introductory

examples for various concepts in condensed matter physics, including entangle-

ment in spin-chains [117–119], matrix product state representations of ground

states [108, 110, 111], bosonic symmetry protected topological (SPT) phases in

one dimension [120,121] and even some aspects of the fractional quantum Hall ef-

fect [122]. Another example of a non-integrable model with a ground state whose

expression is analytically known is the Majumdar-Ghosh model [123], a spin-1/2

Heisenberg chain with an extra fine-tuned next nearest neighbor coupling.

This chapter is organized as follows. In Sec. 1.2, we derive exact expressions

review the spin-1 AKLT model and the construction of its ground state using the

dimer basis. In Sec. 1.3 we introduce the concept of exact states, i.e. eigenstates

having an analytic closed-form expression. We discuss a numerical approach based

on the rank of the reduced density matrix to track these states in exact diago-

nalization studies. We show an extensive numerical study of the spectrum of the

spin-1 AKLT chain, listing all the exact states up to 16 spins. We then proceed to

derive the analytical expressions for all the states. We first consider the low energy

states in Sec. 1.4, recovering the two Arovas states. [124] In Sec. 1.5, we derive the

tower of states, a series of spin-2 magnon excitations on top of the ground state,

ranging from the ground state to the highest energy state and present evidence

that shows that their position in the bulk of the energy spectrum. To show that

22

our approach is valid beyond the spin-1 AKLT chain, we discuss its generalization

to higher integer spin-S in Sec. 1.6, obtaining the analytical expression of all the

exact states that we numerically observe to have a low entanglement rank.

1.2 The Spin-1 AKLT Model

1.2.1 Hamiltonian

The spin-1 AKLT Hamiltonian is defined as a sum projectors that projects two

nearest neighbor spins onto spin 2 [54,114]. Denoting the projector of two spin-1s

on sites i and j onto total spin 2 as P (2,1)ij , the AKLT Hamiltonian for a chain of

length L with periodic boundary conditions (i.e., L+ 1 ≡ 1) simply reads

H =L∑i=1

P(2,1)i,i+1, P

(2,1)ij =

1

24(Si + Sj)

2((Si + Sj)2 − 2). (1.1)

Simplifying the expression Eq. (1.1), the AKLT Hamiltonian can be written in a

more familiar form as

H =L∑i=1

(1

3+

1

2Si · Si+1 +

1

6(Si · Si+1)

2

). (1.2)

The AKLT Hamiltonian Eq. (1.2) has many symmetries. In particular, it pos-

sesses SU(2), translation and inversion (reflection about a bond i, i + 1) sym-

metries. Here, we associate the following quantum numbers to all the eigenstates

of the Hamiltonian: s for the total spin, Sz for the projection of the total spin

along the z direction, and momentum k (quantized in integer multiples of 2π/L)

for the translation symmetry. Furthermore, inversion symmetry maps states with

23

momentum k to states with momentum −k. Hence the eigenstates with momen-

tum k = 0, π can be labelled with a quantum number I = ±1 corresponding to

inversion symmetry. Similarly, eigenstates with Sz = 0 can be labelled by another

quantum number Pz = ±1 corresponding to the spin-flip (Sz → −Sz) symmetry,

which is a part of the SU(2) symmetry.

In spite of these symmetries, the AKLT Hamiltonian is non-integrable. Indeed

the energy levels of eigenstates with a fixed set of quantum numbers corresponding

to different symmetries do show level repulsion. In Fig. 1.1, we plot the energy

level spacing statistics of a typical quantum number sector of the AKLT model,

and we find that the level spacing distribution is close to that of a Gaussian

Orthogonal Ensemble (GOE).

1.2.2 Ground state

The beauty of the AKLT model is that despite its lack of integrability, the ground

state can be constructed explicitly [54]. To do this, we write each spin-1 as two

symmetrized spin-1/2 degrees of freedom that can either have Sz = +1/2 or

Sz = −1/2. Thus, two nearest neighbor spins consist of four spin-1/2 degrees of

freedom. If a singlet is formed between two of them as in Fig. 1.2, the remaining

two spin-1/2’s can form at most a spin-1 configuration, meaning that the projector

P(2,1)i,i+1 annihilates such a configuration. The cartoon picture of the ground state

|G〉 of energy E = 0 with periodic boundary conditions is shown in Fig. 1.2a. With

open boundary conditions, there are two spin-1/2 degrees of freedom at each edge

that are not bound into singlets, so-called dangling spins. These fractionalized

degrees of freedom, i.e. half-odd integer spins in a model that contained only

integer spins, represent the topological nature of the spin-1 Haldane phase. [115]

24

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5∆E

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

P(∆E

)

Figure 1.1: Energy level spacing statistics for L = 16 with periodic boundaryconditions in a typical quantum number sector (s, Sz, k, I, Pz) = (4, 0, 0,−1, 1)that has a Hilbert space dimension 26429. ∆E is the level spacing between adja-cent energy levels after a mapping of the energy spectrum to produce a constantdensity of states. P (∆E) is the distribution of the level spacings. The peak ofthe distribution at non-zero ∆E indicates level repulsion. The green curve is theGOE distribution. The mean ratio of adjacent level spacings is 〈r〉 ≈ 0.5316, closeto the GOE value of 〈r〉 ≈ 0.5295. 3

c c c c c c c cc c c c c c c cm m m m m m m m- - - - - - - - -

FIG. 2. AKLT Ground state. The big circles are physicalspin-1 s and the smaller circles within the spin-1 s are spin-1/2Schwinger bosons. Symmetric combinations of the Schwingerbosons on each site form the physical spin-1. The lines joiningthe Schwinger bosons represent singlets. |Gi with periodicboundary conditions

B. Ground state

The beauty of the AKLT model is that despite itslack of integrability, the ground state can be constructedexplicitly.54 To do this, we write each spin-1 as two sym-metrized spin-1/2 degrees of freedom that can either haveSz = +1/2 or Sz = 1/2. Thus, two nearest neighborspins consist of four spin-1/2 degrees of freedom. If asinglet is formed between two of them as in Fig. 2, theremaining two spin-1/2’s can form at most a spin-1 con-

figuration, meaning that the projector P (2,1)i,i+1

annihilatessuch a configuration. The cartoon picture of the groundstate |Gi of energy E = 0 with periodic boundary con-ditions is shown in Fig. 2a. With open boundary condi-tions, there are two spin-1/2 degrees of freedom at eachedge that are not bound into singlets, so-called dangling

spins. These fractionalized degrees of freedom, i.e. half-odd integer spins in a model that contained only inte-ger spins, represent the topological nature of the spin-1Haldane phase.29 The ground state for open boundaryconditions is shown in Appendix J 1.

In this paper, we mainly work with periodic boundaryconditions (PBC), although we comment on open bound-ary conditions (OBC) in Appendix J. In this case, it hasbeen shown that |Gi is the unique ground state (withenergy 0) of the Hamiltonian Eq. (3).27,28 The ground-state |Gi is separated from the excitation spectrum by anenergy gap27 (the “Haldane gap”), which for the AKLTmodel can be bounded from below. The AKLT groundstate shown in Fig. 2a can be described more rigorouslyusing a dimer (a singlet) basis. Since the spin-1/2 de-grees of freedom on each site are symmetrized, it is con-venient to introduce Schwinger bosons, i.e. bosonic cre-ation (annihilation) operators a†i (ai) and b†i (bi) for theSz = +1/2 (") and Sz = 1/2 (#) spin-1/2 degrees offreedom, respectively. Any wavefunction written in termsof Schwinger bosons on site i can be converted to thenormalized spin-1 basis on a site i (|1ii , |0ii , |-1ii corre-sponding to Sz = +1, 0, -1) using the dictionary:

|1ii =(a†i )

2

p2

|ii , |0ii = a†i b†i |ii , |-1ii =

(b†i )2

p2

|ii(4)

where |ii is the local vacuum defined by the kernel ofthe boson annihilation operators of site i, i.e. ai |ii =0, bi |ii = 0. Since there are two spin-1/2 degrees of

freedom on each site, the number operator Ni = a†iai +

b†i bi has the constraint Ni | i = 2 | i where | i is anyconfiguration of spin-1s. To describe dimers, one couldthen define a dimer creation operator that forms singletsbetween the bosons on di↵erent sites as

c†ij = a†i b†j a†jb

†i . (5)

The complete algebra of dimers and Schwinger bosons isgiven in Appendix B.The spin-1/2 Schwinger boson creation and annihila-

tion operators can be related to the spin-1 operators by

Sz =1

2(a†iai b†i bi)

S+ = a†i bi

S = b†iai. (6)

In this notation, the operator ~Si. ~Sj can be written as

~Si · ~Sj =1

2(S+

i Sj + S

i S+

j ) + Szi S

zj

= 1

2c†ijcij +

1

4(a†iai + b†i bi)(a

†jaj + b†jbj)

= 1 1

2c†ijcij (7)

where we have made use of the Schwinger boson num-ber constraint on each site. Using Eq. (7), Eq. (3) canbe written in terms of dimer creation and annihilationoperators. In particular, P (2,1)

ij can be written as

P(2,1)ij = 1 5

12c†ijcij +

1

24c†ijcijc

†ijcij

= 1 1

4c†ijcij +

1

24c†ij

2

c2ij (8)

where the expression has been normal ordered usingEq. (B7).With this representation, the unnormalized AKLT

state |Gi of Fig. 2a for periodic boundary conditions canbe written as

|Gi =LY

i=1

c†i,i+1

|i . (9)

Here the vacuum |i is the global vacuum defined as thekernel of all the annihilation operators, i.e. cij |i =ai |i = bi |i = 0, 8 i, j. We also define the normalized

ground state28 g|Gi as

g|Gi = |Gip3L + 3(1)L

. (10)

III. EXACT STATES

The energy spectrum of the spin-1 (and, as we will see,integer spin-S) AKLT model for a finite size chain withperiodic boundary conditions exhibits some remarkable

Figure 1.2: AKLT Ground state. The big circles are physical spin-1 s and thesmaller circles within the spin-1 s are spin-1/2 Schwinger bosons. Symmetric com-binations of the Schwinger bosons on each site form the physical spin-1. The linesjoining the Schwinger bosons represent singlets. |G〉 with periodic boundary con-ditions

In this chapter, we work with periodic boundary conditions (PBC). In this

case, it has been shown that |G〉 is the unique ground state (with energy 0) of

the Hamiltonian Eq. (1.2) [54, 114]. The groundstate |G〉 is separated from the

excitation spectrum by an energy gap [54] (the “Haldane gap”), which for the

AKLT model can be bounded from below. The AKLT ground state shown in

25

Fig. 1.2a can be described more rigorously using a dimer (a singlet) basis. Since

the spin-1/2 degrees of freedom on each site are symmetrized, it is convenient

to introduce Schwinger bosons, i.e. bosonic creation (annihilation) operators a†i(ai) and b†i (bi) for the Sz = +1/2 (↑) and Sz = −1/2 (↓) spin-1/2 degrees of

freedom, respectively. Any wavefunction written in terms of Schwinger bosons on

site i can be converted to the normalized spin-1 basis on a site i (|1〉i , |0〉i , |-1〉icorresponding to Sz = +1, 0, -1) using the dictionary:

|1〉i =(a†i )

2

√2| 0 〉i , |0〉i = a†ib

†i | 0 〉i , |-1〉i =

(b†i )2

√2| 0 〉i (1.3)

where | 0 〉i is the local vacuum defined by the kernel of the boson annihilation

operators of site i, i.e. ai | 0 〉i = 0, bi | 0 〉i = 0. Since there are two spin-1/2 degrees

of freedom on each site, the number operator Ni = a†iai + b†ibi has the constraint

Ni |ψ〉 = 2 |ψ〉 where |ψ〉 is any configuration of spin-1s. To describe dimers, one

could then define a dimer creation operator that forms singlets between the bosons

on different sites as

c†ij = a†ib†j − a

†jb

†i . (1.4)

The complete algebra of dimers and Schwinger bosons is given in Appendix A.1.

The spin-1/2 Schwinger boson creation and annihilation operators can be re-

lated to the spin-1 operators by

Sz =1

2(a†iai − b

†ibi), S+ = a†ibi, S− = b†iai, Si · Sj = 1− 1

2c†ijcij, (1.5)

where we have made use of the Schwinger boson number constraint on each site.

Using Eq. (1.5), Eq. (1.2) can be written in terms of dimer creation and annihi-

26

lation operators. In particular, P (2,1)ij can be written as (after normal ordering)

P(2,1)ij = 1− 1

4c†ijcij +

1

24c†ij

2c2ij. (1.6)

With this representation, the unnormalized AKLT state |G〉 of Fig. 1.2a for

periodic boundary conditions can be written as

|G〉 =L∏i=1

c†i,i+1 | 0 〉 . (1.7)

Here the vacuum | 0 〉 is the global vacuum defined as the kernel of all the annihi-

lation operators, i.e. cij | 0 〉 = ai | 0 〉 = bi | 0 〉 = 0, ∀ i, j.

1.3 Exact States

The energy spectrum of the spin-1 (and, as we will see, integer spin-S) AKLT

model for a finite size chain with periodic boundary conditions exhibits some

remarkable features. Beyond the unique ground state whose energy is 0 (for H

of Eq. (1.2)), there are many other states with rational energies up to machine

precision, some of them seemingly located in the bulk of the spectrum. Moreover,

several of these states are even at integer energies. This observation holds for

chains with a length up to L = 16, the upper numerical limit where we can

compute the full spectrum. In this chapter, we show that states with such rational

energies are not coincidences and we can derive analytical expressions for them

akin to Eq. (1.7) for the ground state. Being exact eigenstates for particular finite

system sizes with a closed analytical expression and rational energy, we dub these

states “exact states”.

27

One could argue that looking for exact states by targeting rational energies is

ad-hoc. Indeed, rescaling the energy by a random positive number or shifting the

ground state energy would scramble this information although simple algorithms

could be devised to recover it. Moreover, in finite precision arithmetic, any number

can be written as a rational number. To hunt for possible exact states, we propose

another approach based on the entanglement spectrum [125].

For any eigenstate |ψ〉 of a spin-S chain, we consider the spatial partition into

two continuous regions A and B with LA spins in A and LB spins in B. We

then construct the reduced density matrix ρA = TrB |ψ〉 〈ψ|. The entanglement

spectrum is the eigenvalue spectrum of − log ρA. Assuming that LA ≤ LB, the

rank of ρA (i.e. the number of levels in the entanglement spectrum) is bounded by

(2S + 1)LA . Unless |ψ〉 has a peculiar structure, this bound is usually saturated.

Most eigenstates (including ground states) of local Hamiltonians saturate this

bound. The fact that the entanglement entropy of the ground state of a gapped

Hamiltonian is not volume law [126] merely means that the ground state of the

system can be approximated by a state with a sparse spectrum [112]. However,

states whose entanglement spectrum is truly sparse and whose number of levels in

the entanglement spectrum do not saturate the bound are special. This includes

the ground state of the AKLT model Eq. (1.7), which has exactly 4 out of (2S +

1)LA levels in its entanglement spectrum irrespective of the length LA.

We propose to use the sparsity of entanglement spectrum, the ratio between

the rank of ρA and its dimension, as a probe to search for exact states. A brute

force approach is thus to numerically compute all the eigenstates for a given

system size and label them with their quantum numbers. We then focus on those

exhibiting an entanglement spectrum sparsity at the largest possible value of LA

28

(the integer part of L/2). For the spin-1 AKLT model, we observe that most of

the states in the bulk of the full energy spectrum have a sparsity close or equal to

1, as expected. However, there are a few eigenstates that have an entanglement

spectrum sparsity less than 5% for the largest system sizes we have computed. For

reasons that are still not fully clear to us, most of these eigenstates coincide with

those having a rational energy. Of course, some of these states are trivially exact

states. An example is the highest excited state of the AKLT Hamiltonian that

has all spin-1s with Sz = 1. It is a product state and its reduced density matrix

has a single eigenvalue. In addition, we can then build several exact spin-wave

excitations with one of two spin-flips on top of the ferromagnetic states, which

are also included in our list of exact eigenstates. But as we show below, many of

these exact states have an interesting non-trivial structure. We list these set of

exact states for system sizes L = 12, L = 14 and L = 16 for the spin-1 case with

all their useful quantum numbers and degeneracies in Table 1.1. The derivation

of their analytic expressions will be detailed in the following sections.

1.4 Exact Low energy Excited states of the Spin-1 AKLT

Model

To obtain expressions for the exact excited states of the AKLT model, we need to

choose a convenient basis to work with. We use the dimer basis that introduced

in Sec. 1.2. Here, the basis states are defined as linearly independent states of

dimer or Schwinger boson creation operators acting on the vacuum | 0 〉. Though

this representation allows an elegant representation of (most of) the exact states

we will discuss, the set of dimer basis states is highly overcomplete and non-

29

orthonormal. To study the AKLT model in the dimer basis, we need to derive rules

for the scattering of basis states upon the action of the Hamiltonian. Since the

Hamiltonian in Eq. (1.6) is normal ordered, it is sufficient to compute the actions

of cmij on the basis states. The actions of cij and P (2,1)ij on various configurations on

dimers along with some useful identities are specified in Appendix A and Fig. 1.12.

We have already exemplified the construction of the ground state in Sec. 1.2.2. We

now focus on two exact low energy excited states, namely the Arovas states [124].

1.4.1 Arovas A state

Consider a configuration of dimers |An〉 defined in Eq. (1.8) with the cartoon

picture of dimers as shown in Fig. 1.3a.

|An〉 =

(n−2∏j=1

c†j,j+1

)c†n−1,n+2(c

†n,n+1)

2

(L∏

j=n+2

c†j,j+1

)| 0 〉 . (1.8)

The Arovas A state is a translation invariant linear superposition of these |An〉

with momentum k = π. Up to a global normalization factor, it is given by

|A〉 =∑

n (−1)n |An〉. The system size is even, greater than 5 sites and we impose

periodic boundary conditions.

For pedagogical purposes, we now show the derivation of this first exact state

beyond the ground state. This exemplifies the mechanism that underlies the

derivation of all these exact states of the AKLT Hamiltonian. The proof relies on

several properties of the dimer basis that are given in Appendix A.2. From the

Hamiltonian Eq. (1.1), we deduce that the only terms in the Hamiltonian that

give a non-vanishing contribution upon action on |An〉 are P (2,1)n−1,n and P

(2,1)n+1,n+2.

30

That is,

H |An〉 = (P(2,1)n−1,n + P

(2,1)n+1,n+2) |An〉 . (1.9)

Using the scattering rules in Fig. 1.12 and the cartoon picture of |An〉, it is easy

to see that

P(2,1)n−1,n |An〉 = |An〉+

1

6|An−1〉+

1

2|G〉+ |Bn〉 (1.10)

where |Bn〉 is shown in Fig. 1.3b and defined as

|Bn〉 =

(n−3∏j=1

c†j,j+1

)c†n−1,nc

†n−2,n+1c

†n−1,n+2c

†n,n+1

(L∏

j=n+2

c†j,j+1

)| 0 〉 , (1.11)

and |G〉 is the unnormalized ground state of Eq. (1.7). As seen in Fig. 1.3a, |An〉

is symmetric under inversion about the mid bond n, n+1. Thus the scattering

terms obtained by P (2,1)n+1,n+2 |An〉 are the same as those in Eq. (1.10), but with all

the terms inverted about the mid bond n, n + 1. Under bond inversion, since

|Bn〉 → |Bn+1〉, |An−1〉 → |An+1〉 and |G〉 → |G〉, we obtain

P(2,1)n+1,n+2 |An〉 = |An〉+

1

6|An+1〉+

1

2|G〉+ 1

2|Bn+1〉 . (1.12)

It is important to note that |Bn〉 transforms in that way because it is symmetric

about a site n, and not about any bond. Combining Eqs. (1.10) and (1.12) with

Eq. (1.9), we obtain

H |An〉 = 2 |An〉+1

6(|An−1〉+ |An+1〉) + |G〉+

1

2(|Bn〉+ |Bn+1〉). (1.13)

It follows that H |A〉 = 2 |A〉 − 13|A〉 = 5

3|A〉. Thus, |A〉 is an exact eigenstate of

the spin-1 AKLT Hamiltonian with energy E = 53

and momentum k = π. The

31

6

c c cc c c c c cc c cc cc cm m m m m m m m- - - - - ----

nn-1 n+1 n+2

(a)

c c c c c c c cc c c c c c c cm m m m m m m m- - - - - - -- -

nn-1 n+1 n+2

(b)

FIG. 4. (a) Arovas A configuration |Ani (b) Scattering termconfiguration |Bni

where |Bni is shown in Fig. 4b and defined as

|Bni =Qn3

j=1

c†j,j+1

c†n1,nc

†n2,n+1

c†n1,n+2

c†n,n+1

QLj=n+2

c†j,j+1

|i (15)

and |Gi is the unnormalized ground state of Eq. (9). Asseen in Fig. 4a, |Ani is symmetric under inversion aboutthe mid bond n, n + 1. Thus the scattering terms ob-

tained by P(2,1)n+1,n+2

|Ani are the same as those in Eq. (14),but with all the terms inverted about the mid bondn, n + 1. Under bond inversion, since |Bni ! |Bn+1

i,|An1

i ! |An+1

i and |Gi ! |Gi, we obtain

P(2,1)n+1,n+2

|Ani = |Ani+1

6|An+1

i+ 1

2|Gi+ 1

2|Bn+1

i .(16)

It is important to note that |Bni transforms in that waybecause it is symmetric about a site n, and not aboutany bond. Combining Eqs. (14) and (16) with Eq. (13),we obtain

H |Ani = 2 |Ani+ 1

6

(|An1

i+ |An+1

i) + |Gi+ 1

2

(|Bni+ |Bn+1

i). (17)

It follows that

H |Ai = 2 |Ai 1

3|Ai = 5

3|Ai . (18)

Thus, |Ai is an exact eigenstate of the spin-1 AKLTHamiltonian with energy E = 5

3

and momentum k = .The crucial part of the exactness of this state lies in thefact that |Bni and |Bn+1

i appeared in Eq. (17) with equalweight and they could be cancelled o↵ by momentum su-perposition. However, the cancellation would not hold ifL were not even (since we need k = ) or for open bound-ary conditions (because the edge scattering terms wouldnot cancel). Moreover, |Ai appears only for L 6 be-cause for L = 4, the scattering equation of |Ani Eq. (17)would no longer be the same due to boundary conditions.

Using Eqs. (D2) and (7), it can be seen that f|Ai, thenormalized version of |Ai, can be rewritten as49

f|Ai = 3p2L

LX

n=1

(1)n~Sn · ~Sn+1

g|Gi. (19)

Thus, we have an exact eigenstate with E =

53, k =

and s = 0 (and Pz = +1, I = +1). This state is an exact

c c c c c c c c cc c c c c c c c cm m m m m m m m m- - - - - - - -- -

nn-1 n+1 n+2n-2

(a)

c c c cc c c c c c cc c cc cc cm m m m m m m m m- - - - - - ----

nn-2 n-1 n+1 n+2

(b)

c c c c c c c c cc c c c c c c c cm m m m m m m m m- - - - - - --

--

nn-1 n+1 n+2n-2

(c)

c c c c c c c c cc c c c c c c c cm m m m m m m m m- - - - - - - --

-nn-1 n+1 n+2n-2

(d)

c c cc c c c cc cc c c cc c c cm m m m m m m m m- - - - -- -- --

nn-1 n+1 n+2n-2

(e)

FIG. 5. (a) Arovas B configuration |Bni. (b) Scattering term|Ani. (c) Scattering term |Cni. (d) Scattering term |Dni. (e)Scattering term |Eni.

example of the Single-Mode Approximation (SMA). Thiscan be viewed as a magnon with form factor sin(k) asevident from the momentum space expression of the state

f|Ai = 3p2L

X

k

sin(k)~Sk · ~Skg|Gi (20)

where ~Sk is the Fourier transform of ~Sn.

B. Arovas B state

Similar to the Arovas A state built from the |Anis,another exact state can be constructed from |Bnis ofEq. (15). The Arovas B state then reads

|Bi =X

n

(1)n |Bni (21)

up to a global normalization factor.We consider the configuration |Bni defined in Eq. (15)

and also shown in Fig. 5a. Since the only nearest neigh-bor bonds that do not have dimers between them aren 2, n 1 and n+ 1, n+ 2, analogous to Eq. (13),we can write

H |Bni = (P (2,1)n2,n1

+ P(2,1)n+1,n+2

) |Bni . (22)

The scattering terms of obtained upon action of

P(2,1)n2,n1

, by using rules of Eq. (D7) are

P(2,1)n2,n1

|Bni = |Bni+1

4( |Gi+ |An1

i

+ |Cn1

i+ |Dn1

i)

+1

24( |An2

i+ |En1

i) (23)

Figure 1.3: (a) Arovas A configuration |An〉 (b) Scattering term configuration|Bn〉

crucial part of the exactness of this state lies in the fact that |Bn〉 and |Bn+1〉

appeared in Eq. (1.13) with equal weight and they could be cancelled off by

momentum superposition. However, the cancellation would not hold if L were

not even (since we need k = π) or for open boundary conditions (because the

edge scattering terms would not cancel). Moreover, |A〉 appears only for L ≥ 6

because for L = 4, the scattering equation of |An〉 Eq. (1.13) would no longer be

the same due to boundary conditions.

Using Eqs. (A.9) and (1.5), it can be seen that |A〉, the normalized version of

|A〉, can be rewritten as [124]

|A〉 = 3√2L

L∑n=1

(−1)nSn · Sn+1|G〉. (1.14)

Thus, we have an exact eigenstate with E = 53, k = π and s = 0 (and Pz = +1,

I = +1). This state is an exact example of the Single-Mode Approximation

(SMA). This can be viewed as a magnon with form factor sin(k) as evident from

the momentum space expression of the state

|A〉 = 3√2L

∑k

sin(k)Sk · Sπ−k |G〉 (1.15)

32

6

c c cc c c c c cc c cc cc cm m m m m m m m- - - - - ----

nn-1 n+1 n+2

(a)

c c c c c c c cc c c c c c c cm m m m m m m m- - - - - - -- -

nn-1 n+1 n+2

(b)

FIG. 4. (a) Arovas A configuration |Ani (b) Scattering termconfiguration |Bni

where |Bni is shown in Fig. 4b and defined as

|Bni =Qn3

j=1

c†j,j+1

c†n1,nc

†n2,n+1

c†n1,n+2

c†n,n+1

QLj=n+2

c†j,j+1

|i (15)

and |Gi is the unnormalized ground state of Eq. (9). Asseen in Fig. 4a, |Ani is symmetric under inversion aboutthe mid bond n, n + 1. Thus the scattering terms ob-

tained by P(2,1)n+1,n+2

|Ani are the same as those in Eq. (14),but with all the terms inverted about the mid bondn, n + 1. Under bond inversion, since |Bni ! |Bn+1

i,|An1

i ! |An+1

i and |Gi ! |Gi, we obtain

P(2,1)n+1,n+2

|Ani = |Ani+1

6|An+1

i+ 1

2|Gi+ 1

2|Bn+1

i .(16)

It is important to note that |Bni transforms in that waybecause it is symmetric about a site n, and not aboutany bond. Combining Eqs. (14) and (16) with Eq. (13),we obtain

H |Ani = 2 |Ani+ 1

6

(|An1

i+ |An+1

i) + |Gi+ 1

2

(|Bni+ |Bn+1

i). (17)

It follows that

H |Ai = 2 |Ai 1

3|Ai = 5

3|Ai . (18)

Thus, |Ai is an exact eigenstate of the spin-1 AKLTHamiltonian with energy E = 5

3

and momentum k = .The crucial part of the exactness of this state lies in thefact that |Bni and |Bn+1

i appeared in Eq. (17) with equalweight and they could be cancelled o↵ by momentum su-perposition. However, the cancellation would not hold ifL were not even (since we need k = ) or for open bound-ary conditions (because the edge scattering terms wouldnot cancel). Moreover, |Ai appears only for L 6 be-cause for L = 4, the scattering equation of |Ani Eq. (17)would no longer be the same due to boundary conditions.

Using Eqs. (D2) and (7), it can be seen that f|Ai, thenormalized version of |Ai, can be rewritten as49

f|Ai = 3p2L

LX

n=1

(1)n~Sn · ~Sn+1

g|Gi. (19)

Thus, we have an exact eigenstate with E =

53, k =

and s = 0 (and Pz = +1, I = +1). This state is an exact

c c c c c c c c cc c c c c c c c cm m m m m m m m m- - - - - - - -- -

nn-1 n+1 n+2n-2

(a)

c c c cc c c c c c cc c cc cc cm m m m m m m m m- - - - - - ----

nn-2 n-1 n+1 n+2

(b)

c c c c c c c c cc c c c c c c c cm m m m m m m m m- - - - - - --

--

nn-1 n+1 n+2n-2

(c)

c c c c c c c c cc c c c c c c c cm m m m m m m m m- - - - - - - --

-nn-1 n+1 n+2n-2

(d)

c c cc c c c cc cc c c cc c c cm m m m m m m m m- - - - -- -- --

nn-1 n+1 n+2n-2

(e)

FIG. 5. (a) Arovas B configuration |Bni. (b) Scattering term|Ani. (c) Scattering term |Cni. (d) Scattering term |Dni. (e)Scattering term |Eni.

example of the Single-Mode Approximation (SMA). Thiscan be viewed as a magnon with form factor sin(k) asevident from the momentum space expression of the state

f|Ai = 3p2L

X

k

sin(k)~Sk · ~Skg|Gi (20)

where ~Sk is the Fourier transform of ~Sn.

B. Arovas B state

Similar to the Arovas A state built from the |Anis,another exact state can be constructed from |Bnis ofEq. (15). The Arovas B state then reads

|Bi =X

n

(1)n |Bni (21)

up to a global normalization factor.We consider the configuration |Bni defined in Eq. (15)

and also shown in Fig. 5a. Since the only nearest neigh-bor bonds that do not have dimers between them aren 2, n 1 and n+ 1, n+ 2, analogous to Eq. (13),we can write

H |Bni = (P (2,1)n2,n1

+ P(2,1)n+1,n+2

) |Bni . (22)

The scattering terms of obtained upon action of

P(2,1)n2,n1

, by using rules of Eq. (D7) are

P(2,1)n2,n1

|Bni = |Bni+1

4( |Gi+ |An1

i

+ |Cn1

i+ |Dn1

i)

+1

24( |An2

i+ |En1

i) (23)

Figure 1.4: (a) Arovas B configuration |Bn〉. (b) Scattering term |An〉. (c) Scat-tering term |Cn〉. (d) Scattering term |Dn〉. (e) Scattering term |En〉.

where Sk is the Fourier transform of Sn.

1.4.2 Arovas B state

Similar to the Arovas A state built from the |An〉s, another exact state can be

constructed from |Bn〉s of Eq. (1.11). The Arovas B state then reads |B〉 =∑n (−1)n |Bn〉 up to a global normalization factor. We consider the configuration

|Bn〉 defined in Eq. (1.11) and also shown in Fig. 1.4a. Since the only nearest

neighbor bonds that do not have dimers between them are n − 2, n − 1 and

n+ 1, n+ 2, analogous to Eq. (1.9), we can write

H |Bn〉 = (P(2,1)n−2,n−1 + P

(2,1)n+1,n+2) |Bn〉 . (1.16)

33

The scattering terms of obtained upon action of P (2,1)n−2,n−1, by using rules in

Fig. 1.12 are

P(2,1)n−2,n−1 |Bn〉 = |Bn〉+

1

4(− |G〉+|An−1〉+|Cn−1〉+|Dn−1〉)+

1

24(− |An−2〉+|En−1〉),

(1.17)

where |An〉, |Cn〉, |Dn〉, |En〉 are defined according to the cartoon pictures in

Figs. 1.4b, 1.4c, 1.4d and 1.4e. Since the scattering configurations are all symmet-

ric terms that are symmetric under inversion about a bond n, n + 1, whereas

|Bn〉 is symmetric under inversion about site n, the action of P (2,1)n+1,n+2 changes

|An−1〉 → |An〉, |Cn−1〉 → |Cn〉, |Dn−1〉 → |Dn〉 and |En−1〉 → |En〉. With this

property, for L even with periodic boundary conditions, we obtain H |B〉 = 2 |B〉.

Thus, we have an exact state with E = 2, k = π and s = 0 (and Pz = +1,

I = −1). Again the key ingredient for this derivation was the fact that the scat-

tering terms were symmetric and have the opposite (site/bond) symmetry. As

for |A〉, it is not hard to see that this result would not hold for open boundary

conditions or for odd L. Moreover, |B〉 appears only for L ≥ 8: we need L ≥ 5

to define |Bn〉 and for L = 6 with periodic boundary conditions, the state itself

vanishes.

To formulate |B〉 within the SMA, we need to note that we obtain |Bn〉 by

acting the c†c term of the projector in Eq. (1.6) on |An〉. With this observation,

along with identities of Eq. (A.9), we obtain the normalized eigenstate

|B〉 = 1

4

√27

22L

L∑n=1

(−1)nSn−1 · Sn, Sn · Sn+1|G〉. (1.18)

34

7

where |Ani, |Cni, |Dni, |Eni are defined according to thecartoon pictures in Figs. 5b, 5c, 5d and 5e. A few typo-graphical errors of Eq. (8) in Ref. [49] including the omis-sion of the scattering term |En1

i have been correctedhere in Eq. (24). Since the scattering configurations areall symmetric terms that are symmetric under inversionabout a bond n, n+ 1, whereas |Bni is symmetric un-

der inversion about site n, the action of P (2,1)n+1,n+2

changes|An1

i ! |Ani, |Cn1

i ! |Cni, |Dn1

i ! |Dni and|En1

i ! |Eni. The action of the Hamiltonian on |Bnithus reads

H |Bni = 2 |Bni 1

2|Gi+ 1

4(|An1

i+ |Ani)

+1

4(|Cn1

i+ |Cni) +1

4(|Dn1

i+ |Dni) (24)

1

24(|An2

i+ |An+1

i) + 1

24(|En1

i+ |Eni).

With this property, for L even with periodic boundaryconditions, we obtain

H |Bi = 2 |Bi . (25)

Thus, we have an exact state with E = 2, k = ands = 0 (and Pz = +1, I = 1). Again the key ingredientfor this derivation was the fact that the scattering termswere symmetric and have the opposite (site/bond) sym-metry. As for |Ai, it is not hard to see that this resultwould not hold for open boundary conditions or for oddL. Moreover, |Bi appears only for L 8: we need L 5to define |Bni and for L = 6 with periodic boundaryconditions, the state itself vanishes.To formulate |Bi within the SMA, we need to note that

we obtain |Bni by acting the c†c term of the projector inEq. (8) on |Ani. With this observation, along with iden-tities of Eq. (D3), we obtain the normalized eigenstate49

g|Bi =q

27

22L

PLn=1

(1)n~Sn · ~Sn+1

1

2

(~Sn1

· ~Sn)(~Sn · ~Sn+1

)g|Gi (26)

Eq. (26) can also be written as a Hermitian operator onthe ground state:

g|Bi = 1

4

q27

22L

PLn=1

(1)n~Sn1

· ~Sn, ~Sn · ~Sn+1

g|Gi

(27)

where , denotes the anti-commutator.One might wonder if the pattern of exactness might

continue for other dimer configurations such as |Cni,|Dni or a combination of the two. However, in such cases,the scattering terms are no longer inversion symmetric,and hence do not appear in pairs, precluding cancella-tions at k = . For the Arovas states considered here,the scattering terms appeared in pairs due to the pres-ence of only two non-vanishing projectors on the state,forcing the exact state to have a momentum k = .

c c c c c c cc c c c c c cm m m m m m m- - - - - -6 66 6nn-1 n+1

(a)

c c c c c c cc c c c c c cm m m m m m m- - - - --6 66 6nn-1 n+1

(b)

FIG. 6. The two configurations that appear in the derivationof the spin-2 magnon state (a) The spin-2 magnon state |Mni.(b) Scattering state |Nni.

V. MID-SPECTRUM EXACT STATES

We now move on to the study of non-singlet states,i.e. states with a total spin s 6= 0. Since the AKLTHamiltonian Eq. (1) is SU(2) symmetric, it is sucientto consider the highest weight state of each multipletof spin s. The entire multiplet of 2s + 1 states can beobtained by repeated application of S =

PLn=1

Sn on

the highest weight state.

A. Spin-2 magnon state

We start with a configuration with the cartoon pic-ture as shown in Fig. 6a. This particular configurationis used because of the rule Eq. (D23) of Fig. 27, derivedin Appendix D. It shows a fairly simply and symmet-ric scattering process. We define |Mni as a quasiparticlewith no dimers around site n as

|Mni =n2Y

j=1

c†j,j+1

a†n1

(a†n)2a†n+1

LY

j=n+1

c†j,j+1

|i . (28)

From the cartoon picture Fig. 6a, it is clear that the onlyprojectors in the Hamiltonian that do not vanish on |Mniare P

(2,1)n1,n and P

(2,1)n,n+1

. Using Eq. (D23),

P(2,1)n1,n |Mni = |Mni+

1

2|Nn1

i . (29)

where |Nni is shown in Fig. 6b. Since |Nni is bond in-version symmetric under bond n, n+1 whereas |Mni issite inversion symmetric about site n (they have oppositetypes of symmetries), the action of the Hamiltonian on|Mni reads

H |Mni = 2 |Mni+1

2(|Nn1

i+ |Nni) (30)

and one can use the k = superposition to remove the|Nni states. The translation invariant state is thus

|S2

i =LX

n=1

(1)n |Mni. (31)

With L even and periodic boundary conditions,

H |S2

i = 2 |S2

i . (32)

Figure 1.5: The two configurations that appear in the derivation of the spin-2magnon state (a) The spin-2 magnon state |Mn〉. (b) Scattering state |Nn〉.

1.5 Mid-Spectrum Exact States

We now move on to the study of non-singlet states, i.e. states with a total spin s 6=

0. Since the AKLT Hamiltonian Eq. (1.1) is SU(2) symmetric, it is sufficient to

consider the highest weight state of each multiplet of spin s. The entire multiplet

of 2s+1 states can be obtained by repeated application of S− =∑L

n=1 S−n on the

highest weight state.

1.5.1 Spin-2 magnon state

We start with a configuration with the cartoon picture as shown in Fig. 1.5a. We

define |Mn〉 as a quasiparticle with no dimers around site n as

|Mn〉 =n−2∏j=1

c†j,j+1a†n−1(a

†n)

2a†n+1

L∏j=n+1

c†j,j+1 | 0 〉 . (1.19)

From the cartoon picture Fig. 1.5a, it is clear that the only projectors in the

Hamiltonian that do not vanish on |Mn〉 are P (2,1)n−1,n and P

(2,1)n,n+1. Using the rules

in Fig. 1.12,

P(2,1)n−1,n |Mn〉 = |Mn〉+

1

2|Nn−1〉 . (1.20)

35

where |Nn〉 is shown in Fig. 1.5b. Since |Nn〉 is bond inversion symmetric under

bond n, n+1 whereas |Mn〉 is site inversion symmetric about site n (they have

opposite types of symmetries), the action of the Hamiltonian on |Mn〉 reads

H |Mn〉 = 2 |Mn〉+1

2(|Nn−1〉+ |Nn〉) (1.21)

and one can use the k = π superposition to remove the |Nn〉 states. The trans-

lation invariant state is thus |S2〉 =∑L

n=1 (−1)n |Mn〉. With L even and periodic

boundary conditions, we obtain H |S2〉 = 2 |S2〉. Thus we have an exact multiplet

of states with s = 2 (4 a†is in the state), E = 2, k = π.

This state can again be written as an SMA with a spin 2 magnon. From

Fig. 1.5a, we immediately see that |Mn〉 = −(S+n2/2) |G〉, that is, the spin on site

n is forced to have Sz = 1. Thus, including the normalization factor, the full state

can be written as

|S2〉 =√

3

4L

L∑n=1

(−1)nS+n2|G〉. (1.22)

This state exists for all L even and L ≥ 4. The entire multiplet of states with

different Sz can be obtained from |S2〉 by applying the S− operator. Note that an

exact spin-2 magnon state can be constructed similarly for the AKLT Hamiltonian

with open boundary conditions as well [55].

1.5.2 Tower of states

We now denote the spin-2 magnon “creation” operator as P =∑L

n=1 (−1)nS+n2.

The state P2 |G〉 is a state with two of the k = π spin-2 magnons dispersing on the

chain. The correct basis state is proportional to S+n2S+2

n+m |G〉, with n+m defined

modulo L, containing two magnons (S+n4 |G〉 = 0 for m = 0). It is convenient to

36

8

c c c c c c c c cc c c c c c c c cm m m m m m m m m- - - - 6 66 6- -6 66 6nn-1 n+1 n+4n+3 n+5

(a)

c c c c c c c c cc c c c c c c c cm m m m m m m m m- - - 6 66 6- -6 66 6-nn-1 n+1 n+4n+3 n+5

(b)

c c c c c c c c cc c c c c c c c cm m m m m m m m m- - - -6 66 6 - -6 66 6nn-1 n+1 n+3n+2

(c)

c c c c c c c c cc c c c c c c c cm m m m m m m m m- - - -6 66 6 - -6 66 6nn-1 n+1 n+3n+2

(d)

FIG. 7. (a) Two magnon configuraton |Mn,Mn+4i. (b)Scattering state |Nn,Mn+4i. (c) Two magnon configura-tion |Mn,Mn+2i. (d) Two magnon scattering configuration|Mn, Nn+2i

Thus we have an exact multiplet of states with s = 2 (4a†i s in the state), E = 2, k = .This state can again be written as an SMA with a

spin 2 magnon. From Fig. 6a, we immediately see that|Mni = (S+

n2

/2) |Gi, that is, the spin on site n is forcedto have Sz = 1. Thus, including the normalization factor,the full state can be written as

g|S2

i =r

3

4L

LX

n=1

(1)nS+

n2g|Gi. (33)

This state exists for all L even and L 4. In terms ofmomentum space operators, the state has the expression

g|S2

i =r

3

4L

X

k

S+

k S+

kg|Gi. (34)

The entire multiplet of states with di↵erent Sz can beobtained from |S

2

i by applying the S operator.An exact spin-2 magnon state can be constructed sim-

ilarly for the AKLT Hamiltonian with open boundary

conditions too, see Appendix J 2.

B. Tower of states

We now denote the spin-2 magnon “creation” oper-ator as P =

PLn=1

(1)nS+

n2

. The state P2 |Gi is astate with two of the k = spin-2 magnons dispers-ing on the chain. The correct basis state is proportionalto S+

n2

S+

2

n+m |Gi, with n + m defined modulo L, con-

taining two magnons (S+

n4 |Gi = 0 for m = 0). It is

convenient to denote this basis state |Mn,Mn+mi, n andn +m denoting the position of the magnons. Similarly,basis states |Mn, Nn+mi can be defined as the config-uration with the spin-2 magnon at position m and thescattering magnon of Fig. 6b at position n + m. SinceS+

n2 |Gi annihilates all spin configurations of |Gi that do

not have Sz = 1 on site n, it follows that Sz 6= 1 onsite n+1 because the sites n and n+1 share a singlet in

|Gi. Thus, (S+

n )2(S+

n+1

)2 |Gi = 0. For other m, there aretwo possibilities. If 3 m L 3, we can write downthe action of the Hamiltonian on the two magnon basisstates using Eq. (30) as

H |Mn,Mn+mi = 4 |Mn,Mn+mi+ 1

2(|Nn1

,Mn+mi

+ |Nn,Mn+mi+ |Mn, Nn+m1

i+ |Mn, Nn+mi) .(35)

An example of such a state and its scattering configu-ration are shown in Fig. 7a and Fig. 7b respectively. Ifm = 2 or m = L 2, the two magnons fuse to form onespin-4 magnon, as shown in Fig. 7c. Using Eq. (D23),the action of the Hamiltonian can be written as

H |Mn,Mn+2

i = 4 |Mn,Mn+2

i

+1

2(|Nn1

,Mn+2

i+ |Mn, Nn+2

i).(36)

In terms of these basis states, the translation invariantstate comprising of two spin-2 magnons on the AKLTchain is

|S4

i =LX

n=1

L2X

m=2

(1)m |Mn,Mn+mi. (37)

Using Eqs. (35) and (36) the action of the Hamiltonianon |S

4

i can be written as

H |S4

i = 4 |S4

i+ 1

2(|Csi+ |Cti). (38)

where we group the scattering terms into two: |Csi aris-ing from basis states in Eq. (37) of the type Fig. 7a, (i.e.magnons are separated, 3 m L 3) and |Cti arisingfrom basis states of the type Fig. 7c (i.e. magnons arenext to each other, m = 2, L 2). Using Eqs. (35), (36)and (37), |Csi and |Cti can be simplified to

|Csi =LX

n=1

L3X

m=3

(1)m(|Nn1

,Mn+mi+ |Nn,Mn+mi

+ |Mn, Nn+m1

i+ |Mn, Nn+mi) (39)

|Cti =LX

n=1

(|Nn,Mn+L2

i+ |Nn,Mn+3

i

+ |Mn, Nn+2

i+ |Mn, Nn+L3

)i . (40)

In Eq. (39), considering the summation over just the firstscattering term and relabelling the summation indicesconsidering L even, we obtain

PLn=1

PL3

m=3

(1)m |Nn1

,Mn+mi=

PLn=1

PL2

m=4

(1)m |Nn,Mn+mi. (41)

Similarly for the third scattering term in Eq. (39), weobtain

PLn=1

PL3

m=3

(1)m |Mn, Nn+m1

i=

PLn=1

PL4

m=2

(1)m |Mn, Nn+mi. (42)

Figure 1.6: (a) Two magnon configuraton |Mn,Mn+4〉. (b) Scattering state|Nn,Mn+4〉. (c) Two magnon configuration |Mn,Mn+2〉. (d) Two magnon scatter-ing configuration |Mn, Nn+2〉

denote this basis state |Mn,Mn+m〉, n and n + m denoting the position of the

magnons. Similarly, basis states |Mn, Nn+m〉 can be defined as the configuration

with the spin-2 magnon at position m and the scattering magnon of Fig. 1.5b at

position n +m. Since S+n2 |G〉 annihilates all spin configurations of |G〉 that do

not have Sz = −1 on site n, it follows that Sz 6= −1 on site n + 1 because the

sites n and n + 1 share a singlet in |G〉. Thus, (S+n )

2(S+n+1)

2 |G〉 = 0. For other

m, there are two possibilities. If 3 ≤ m ≤ L− 3, we can write down the action of

the Hamiltonian on the two magnon basis states using Eq. (1.21) as

H |Mn,Mn+m〉 = 4 |Mn,Mn+m〉

+1

2(|Nn−1,Mn+m〉+ |Nn,Mn+m〉+ |Mn, Nn+m−1〉+ |Mn, Nn+m〉) .

(1.23)

An example of such a state and its scattering configuration are shown in Fig. 1.6a

and Fig. 1.6b respectively. If m = 2 or m = L − 2, the two magnons fuse to

form one spin-4 magnon, as shown in Fig. 1.6c. Using Fig. 1.12, the action of the

37

Hamiltonian can be written as

H |Mn,Mn+2〉 = 4 |Mn,Mn+2〉+1

2(|Nn−1,Mn+2〉+ |Mn, Nn+2〉). (1.24)

In terms of these basis states, the translation invariant state comprising of two

spin-2 magnons on the AKLT chain is

|S4〉 =L∑n=1

L−2∑m=2

(−1)m |Mn,Mn+m〉. (1.25)

Using Eqs. (1.23) and (1.24) the action of the Hamiltonian on |S4〉 can be written

as

H |S4〉 = 4 |S4〉+1

2(|Cs〉+ |Ct〉). (1.26)

where we group the scattering terms into two: |Cs〉 arising from basis states in

Eq. (1.25) of the type Fig. 1.6a, (i.e. magnons are separated, 3 ≤ m ≤ L− 3) and

|Ct〉 arising from basis states of the type Fig. 1.6c (i.e. magnons are next to each

other, m = 2, L − 2). Using Eqs. (1.23), (1.24) and (1.25), |Cs〉 and |Ct〉 can be

simplified to

|Cs〉 =L∑n=1

L−3∑m=3

(−1)m(|Nn−1,Mn+m〉+ |Nn,Mn+m〉+ |Mn, Nn+m−1〉+ |Mn, Nn+m〉)

|Ct〉 =L∑n=1

(|Nn,Mn+L−2〉+ |Nn,Mn+3〉+ |Mn, Nn+2〉+ |Mn, Nn+L−3)〉 . (1.27)

38

9

c c c c c c c c cc c c c c c c c cm m m m m m m m m- - - -6 66 6 - -6 66 6nn-1 n+1 n+3n+2

(a)

c c c c c c c c cc c c c c c c c cm m m m m m m m m- - - 6 66 6- - -6 66 6nn-1 n+1 n+4n+3n+2

(b)

c c c c c c c c cc c c c c c c c cm m m m m m m m m- - - -6 66 6 - -6 66 6nn-1 n+1 n+3n+2

(c)

FIG. 8. (a) A configuration with magnons on sites n andn+ 2 (b) A configuration with magnons on sites n and n+ 3(c) Common scattering configuration

Adding all the terms in Eq. (39) back, we obtain

|Csi = LX

n=1

(|Nn,Mn+L2

i+ |Nn,Mn+3

i

+ |Mn, Nn+2

i+ |Mn, Nn+L3

i). (43)

Using Eqs. (43) and (40), |Csi+ |Cti = 0. Thus,

H |S4

i = 4 |S4

i . (44)

This is an exact state with s = 4, k = 0 and E = 4.As with all the states we have presented, the cancellationhere works only if L is even.This construction can be easily generalized by noting

that the spin-2 magnons on the spin chain behave assolitons. A state with N k = spin-2 magnons reads

|S2N i =

X

lj

(1)PN

j=1

lj |Ml1

,Ml2

. . .MlN i. (45)

As we have seen earlier, S+

2

nS+

2

n+1

|Gi = 0. Hence,in Eq. (45), all configurations |. . . ,Mn,Mn+1

, . . . i van-ish. Thus the set lj satisfies the constraints 1 j N , lj+1

> lj + 1, 1 lj L where additionis defined modulo L. Upon the action of the Hamilto-nian on Eq. (45), a term |Ml

1

, . . . ,Mlk . . . ,MlN i scat-ters to |Ml

1

, . . . , Npk , . . . ,MlN i where pk = lk 1 orpk = lk. From Eqs. (30), (35) and (36), observe thatfor each such scattering term, there always is a uniquedi↵erent term in Eq. (45) |Ml

1

, . . . ,Mqk . . . ,MlN i whereqk = lk 1 or qk = lk + 1 that scatters to the sameterm. For example, a state with spin-2 magnons onseveral sites, including n and n + 2 (Fig. 8a) and an-other state with spin-2 magnons on the same set of sites,except for n + 2 replaced by n + 3 (Fig. 8b), share ascattering term (Fig. 8c). However, for the scatteringterms to cancel, the terms |Ml

1

, . . . ,Mqk , . . . ,MlN i and|Ml

1

, . . . ,Mlk , . . . ,MlN i should have the opposite sign inEq. (45). This is true for the case when lk = l

1

= 1 andqk = lN = L only if L is even. Thus, all the scatteringterms arising from Eq. (45) cancel out and we have anexact state. |S

2N i can also be written as |S2N i = PN |Gi

and has a momentum k = 0 or depending on whetherN is even or odd. Its total spin is s = 2N , and its en-ergy is E = 2N . Since each spin-2 magnon annihilates

FIG. 9. (Color online) Positions of the Sz = 0 states ofthe tower of states within the energy spectrum of their ownquantum number sector plotted against the energy density = E/L for L = 8, 10, 12, 14, 16 with periodic boundaryconditions. Each dot corresponds to a state with N spin-2magnons with N = 0, 1, . . . , L/2 2. The inset shows thedensity of states for L = 16 in the quantum number sector(s, Sz, k, I, Pz) = (8, 0, 0,1, 1). The vertical green line in theinset indicates the position of |S8i.

two dimers from the ground state and L is even, |SLi isa state with E = L, s = L, the highest excited state ofthe model. Thus, |S

2N i is a tower of exact states fromthe ground state to the highest excited state. In termsof the spin operators, the highest weight states of thistower can be written as

|S2N i = N

X

lj

(1)PN

j=1

lj

NY

j=1

(S+

lj)2g|Gi (46)

where N is a normalization factor. Similar to the spin-2magnon, the entire tower of states can be extended tothe AKLT chain with open boundary conditions, seeAppendix J 3.

C. Position in the energy spectrum

In this section we study the positions of the towerof states |S

2N i in the energy spectrum. It isbelieved12,13,22 that all energy eigenstates of non-integrable models that lie in a region of finite densityof states satisfy ETH. This is commonly known strong

ETH.22,23 However, the local density of states can al-ways be changed by tuning the Hamiltonian to allowlevel crossings from states with di↵erent quantum num-bers. To avoid this possibility, we study the position ofthe tower of states in the density of states of their ownquantum number sector defined by (s, Sz, k, I, Pz).

Figure 1.7: (a) A configuration with magnons on sites n and n + 2 (b) A config-uration with magnons on sites n and n+ 3 (c) Common scattering configuration

In the expression for |Cs〉, considering the summation over just the first scattering

term and relabelling the summation indices considering L even, we obtain

L∑n=1

L−3∑m=3

(−1)m |Nn−1,Mn+m〉 = −L∑n=1

L−2∑m=4

(−1)m |Nn,Mn+m〉. (1.28)

Similarly for the third scattering term in |Ct〉 of Eq. (1.27), we obtain

L∑n=1

L−3∑m=3

(−1)m |Mn, Nn+m−1〉 = −L∑n=1

L−4∑m=2

(−1)m |Mn, Nn+m〉. (1.29)

Using these simplifications along with Eq. (1.27), we obtain |Cs〉 + |Ct〉 = 0 and

thus, H |S4〉 = 4 |S4〉. This is an exact state with s = 4, k = 0 and E = 4.

As with all the states we have presented, the cancellation here works only if L is

even.

This construction can be easily generalized by noting that the spin-2 magnons

on the spin chain behave as solitons. A state with N k = π spin-2 magnons reads

|S2N〉 =∑lj

(−1)∑N

j=1 lj |Ml1 ,Ml2 . . .MlN 〉. (1.30)

As we have seen earlier, S+2nS

+2n+1 |G〉 = 0. Hence, in Eq. (1.30), all configurations

39

|. . . ,Mn,Mn+1, . . . 〉 vanish. Thus the set lj satisfies the constraints 1 ≤ j ≤

N , lj+1 > lj + 1, 1 ≤ lj ≤ L where addition is defined modulo L. Upon the

action of the Hamiltonian on Eq. (1.30), a term |Ml1 , . . . ,Mlk . . . ,MlN 〉 scatters to

|Ml1 , . . . , Npk , . . . ,MlN 〉 where pk = lk−1 or pk = lk. From Eqs. (1.21), (1.23) and

(1.24), observe that for each such scattering term, there always is a unique different

term in Eq. (1.30) |Ml1 , . . . ,Mqk . . . ,MlN 〉 where qk = lk − 1 or qk = lk + 1 that

scatters to the same term. For example, a state with spin-2 magnons on several

sites, including n and n + 2 (Fig. 1.7a) and another state with spin-2 magnons

on the same set of sites, except for n + 2 replaced by n + 3 (Fig. 1.7b), share a

scattering term (Fig. 1.7c). However, for the scattering terms to cancel, the terms

|Ml1 , . . . ,Mqk , . . . ,MlN 〉 and |Ml1 , . . . ,Mlk , . . . ,MlN 〉 should have the opposite sign

in Eq. (1.30). This is true for the case when lk = l1 = 1 and qk = lN = L only if

L is even. Thus, all the scattering terms arising from Eq. (1.30) cancel out and

we have an exact state. |S2N〉 can also be written as |S2N〉 = PN |G〉 and has a

momentum k = 0 or π depending on whether N is even or odd. Its total spin is

s = 2N , and its energy is E = 2N . Since each spin-2 magnon annihilates two

dimers from the ground state and L is even, |SL〉 is a state with E = L, s = L,

the highest excited state of the model. Thus, |S2N〉 is a tower of exact states

from the ground state to the highest excited state. In terms of the spin operators,

the highest weight states of this tower can be written as

|S2N〉 = N∑lj

(−1)∑N

j=1 lj

N∏j=1

(S+lj)2 |G〉 (1.31)

where N is a normalization factor. Similar to the spin-2 magnon, the entire

tower of states can be extended to the AKLT chain with open boundary con-

40

0 0.2 0.4 0.6 0.8 1

ε=E/L

0

5

10

15

20

25

30

35

40

Posi

tion in %

L = 8L = 10L = 12L = 14L = 16

0.0 0.2 0.4 0.6 0.8 1.0ε

0

1

2

3

4

ρ (ε)

Figure 1.8: Positions of the Sz = 0 states of the tower of states within the energyspectrum of their own quantum number sector plotted against the energy den-sity ϵ = E/L for L = 8, 10, 12, 14, 16 with periodic boundary conditions. Eachdot corresponds to a state with N spin-2 magnons with N = 0, 1, . . . , L/2 − 2.The inset shows the density of states for L = 16 in the quantum number sector(s, Sz, k, I, Pz) = (8, 0, 0,−1, 1). The vertical green line in the inset indicates theposition of |S8〉.

ditions [55].

1.5.3 Position in the energy spectrum

In this section we study the positions of the tower of states |S2N〉 in the energy

spectrum. It is believed [14, 15, 96] that all energy eigenstates of non-integrable

models that lie in a region of finite density of states satisfy ETH. This is com-

monly known strong ETH. [96,97] However, the local density of states can always

be changed by tuning the Hamiltonian to allow level crossings from states with

different quantum numbers. To avoid this possibility, we study the position of

the tower of states in the density of states of their own quantum number sector

defined by (s, Sz, k, I, Pz).

41

The inset of Fig. 1.8, we show a typical example of the density of states for the

spin-1 AKLT chain for periodic boundary conditions. We focus on system size

L = 16 and the sector defined by the set of quantum numbers (s, Sz, k, I, Pz) =

(8, 0, 0,−1, 1). The quantum numbers are defined in Table 1.1. In this sector lies,

for example, the state of the tower with N = L/4 magnons, i.e. |S2N〉 = |S8〉. As

can be observed, this state is located in a region with finite density of states. Since

any given state of the tower of states has a fixed energy as the thermodynamic

limit (L→∞) is taken, it is natural to expect that such a state would eventually

lie in a region of zero density of states. However, for a fixed L, we have a tower of

states of E = 2N, s = 2N , N ∈ [0, L/2], and hence the number of states and their

energies increase as the system size increases. One could look at the states at a

finite energy density ϵ = E/L and then take the thermodynamic limit (E → ∞,

L→∞). In this limit, we conjecture that some states of the tower lie in the bulk

of the energy spectrum in the thermodynamic limit.

1.5.4 Exact High Energy States

For completeness, we briefly comment the exact states in the upper part of the

energy spectrum shown in Table 1.1. Not all the states presented here are specific

to the AKLT model. Some are eigenstates merely as a consequence of having

SU(2) symmetry and translation invariance. Moreover, it has been shown that

all the states with quantum numbers Sz = L − 1 and Sz = L − 2 (and hence

s = L − 1 and s = L − 2) can be analytically obtained for a general SU(2)

symmetric spin-1 Hamiltonian in the thermodynamic limit [127] and for any even

system size, [128] in spite of the fact that a general SU(2) symmetric spin-1

Hamiltonian is non-integrable. This is due to the fact that the scattering equations

42

of states with Sz = L − 1 and Sz = L − 2 correspond to one and two-body

scattering problems respectively, which are integrable. Indeed, states in these

quantum number sectors do not exhibit level repulsion. However, for Sz = L− 2,

in most cases it is impossible to obtain a closed-form expression of the eigenstates

for a finite system size. The states presented in this section are thus examples

of high energy eigenstates with s = L − 1 and s = L − 2 that have a simple

analytical expression for any finite system in an otherwise completely solvable

quantum number sector. The scattering equation of states with Sz = L − 3

similarly corresponds to a three-magnon scattering problem, that has been solved

partially [127]. However, we have not found any exact states with s = L− 3.

1.6 1D Spin-S AKLT Models with S > 1

AKLT models can be straightforwardly generalized to all dimensions and also

to spins with different Lie algebras [122, 129–131]. In this section, we consider

the generalization to spin-S with S (Sravya) being a positive integer. Such a

model has been studied to explore the Haldane conjecture for S = 2 [132–135].

Particularly, it has been observed that odd integer spin chains are topological (due

to the presence of half-integer dangling spins at the edge) whereas even integer

spin chains are not [133, 135]. The generalization of the AKLT Hamiltonian that

was used [133, 134] is

H(S) =L∑i=1

2S∑J=S+1

αJP(J,S)i,i+1 (1.32)

with αJ ≥ 0 ∀J . As we will see later, the ground state is the same for all the

Hamiltonians of the form Eq. (1.32). However, the entire energy spectrum is not

the same. For example, the ferromagnetic state need not be the unique highest

43

excited state unless

α2S ≥2S−1∑J=S+1

αJ . (1.33)

As we did for the spin-1 AKLT model, we can write Hamiltonian Eq. (1.32) in

terms of spin operators. The most general expression for P (J,S)i,j , is the projector

onto total spin J for two spin-S on sites i and j and can be written in terms of

the spin operators as

P(J,S)i,j =

2S∏s=0,s =J

(2Si · Sj + 2S(S + 1)− s(s+ 1)

J(J + 1)− s(s+ 1)

). (1.34)

Since we are working with spin-S, each spin can be thought to be composed of

2S spin-1/2 Schwinger bosons, with the same algebra described in Appendix A.1.

However, since the number of Schwinger bosons per site changes, Eq. (1.5) also

changes to

Si · Sj = S2 − 1

2c†ijcij (1.35)

Using Eq. (1.35) and Eq. (A.6) for normal ordering, we obtain an expression for

the spin-S AKLT Hamiltonian,

H(S) =∑i

(1 +

2S∑j=1

γj(c†i,i+1)

j(ci,i+1)j

)(1.36)

where the coefficients γj depend on the coefficients αJ in the projectors and the

spin S, a closed form of which we could not obtain for a general S. We have

the freedom to choose (S− 1) αJ coefficients while retaining the standard ground

state but changing the excitation spectrum. By choosing αJ = 1 for all J , we

observe that it is possible to set γj = 0, 1 ≤ j ≤ S − 1. This is the only choice of

44

14c c c c c c c cc c c c c c c cc c c c c c c cc c c c c c c c- - - - - - - - -- - - - - - - - -

FIG. 14. S = 2 AKLT ground state with two dimers betweennearest neighbors. Spin-S AKLT would have S dimers. |2Giwith periodic boundary conditions.

Using Eq. (71) and Eq. (B7) for normal ordering, weobtain an expression for the spin-S AKLT Hamiltonian,

H(S) =X

i

0

@1 +2SX

j=1

j(c†i,i+1

)j(ci,i+1

)j

1

A (72)

where the coecients j depend on the coecients ↵J inthe projectors and the spin S, a closed form of which wecould not obtain for a general S. We have the freedom tochoose (S1) ↵J coecients while retaining the standardground state but changing the excitation spectrum. Bychoosing ↵J = 1 for all J , we observe that it is possible toset j = 0, 1 j S1. This is the only choice of ↵Jthat satisfies the required condition. The Hamiltonian isthen,

H(S) =X

i

0

@1 +2SX

j=S

j(c†i,i+1

)j(ci,i+1

)j

1

A (73)

As we will show later in this section, this choice of coef-ficients is crucial for us to have non-trivial exact states(including a tower of states) in the bulk of the spectrum.Since the algebra of dimers described in Appendix B isindependent of the spin model we are working with, itholds here as well.

A. Lower spectrum states

We start our analysis of the spin-S AKLT model withits ground state. Working in the spin basis, if S dimersare formed between two spin-S (i and i + 1) that havea total of 4S spin-1/2 Schwinger bosons, the maximumspin of both spin-S combined cannot exceed S. Such a

configuration must therefore be annihilated by all P (J,S)

i,i+1

for J > S, and is thus the unique ground state |SGi ofthe Hamiltonian Eq. (68). This can be written as

|SGi =LY

i=1

(c†i,i+1

)S |i . (74)

The cartoon picture for the ground state of the spin-2AKLT model is shown in Fig. 14. The ground state withopen boundary conditions is computed in Appendix J 4.

One might wonder if any low energy s = 0 magnonssimilar to the two Arovas magnon states discussed inSec. IV exist for the spin-S AKLT models. We have usedthe method described in Sec. III to detect exact eigen-states for S = 2 with L 10 and S = 3 with L 8.

c c c c c c cc c c c c c cc c c c c c cc c c c c c c- -- - - - - - - -- -- -

- -

nn-1 n+1 n+2n-2

(a)

c c c c c c cc c c c c c cc c c c c c cc c c c c c c- -- - - -- -

- -- -- -- -

nn-1n-2 n+1 n+2

(b)

c c c c c c cc c c c c c cc c c c c c cc c c c c c c-- - -- - - - --- --

---

nn-1 n+1 n+2n-2

(c)

c c c c c c cc c c c c c cc c c c c c cc c c c c c c- - -- - - --

--- -- --

-

nn-1n-2 n+1 n+2

(d)

c c c c c c cc c c c c c cc c c c c c cc c c c c c c- - - -- - - --- -

- ---

-nn-1 n+1 n+2n-2

(e)

FIG. 15. The spin-2 AKLT configurations (a) |BGni and(b)

B2n

↵. Such configurations form the generalization of the

Arovas B state in the spin-S AKLT model. (c) An example ofa symmetric configuration that appears only in CS

B2

. (d) Anexample of a symmetric configuration that appears in bothCSBG and CS

B2

. (e) An example of a non-symmetric scatteringconfiguration |i.

In both the systems, we find only one singlet exact stateapart from the ground state, and it lies in the quantumnumber sector s = 0, k = , I = 1 for even L, and hasan energy E = 2. Since the Arovas B state in Sec. IVBis in the same sector, we need a similar configurationwith short-ranged dimers. Numerically, we do not findan analogue of the Arovas A state for higher spin AKLTmodels with the chosen Hamiltonian given by Eq. (73)(i.e. with ↵J = 1 8J in Eq. (68)). At the end of this sec-tion, we provide an intuitive explanation as to why thisis the case. In Appendix H 2, we prove that for S = 2,an Arovas A state cannot be obtained for the Hamilto-nian Eq. (73) but an analogue can be constructed withanother suitable choice of coecients ↵J in Eq. (68).To exemplify the derivation of the generalized Arovas

B state, we first focus on the spin-2 AKLT model. Thecorresponding Hamiltonian Eq. (73) reduces to

H(2) =Pi

1 1

84

c†2

i,i+1

c2i,i+1

+ 1

630

c†3

i,i+1

c3i,i+1

1

6720

c†4

i,i+1

c4i,i+1

. (75)

One could form spin-2 basis states by gluing togethertwo copies of spin-1 basis states, along with completelysymmetrizing the spin-1s of the two copies. The spin-2configuration |BGni (Fig. 15a) is formed by gluing oneone spin-1 Arovas B configuration |Bni (Fig. 5a) to thespin-1 ground state |Gi (Fig. 2) and |BGni (Fig. 15b)by gluing two |Bnis together. The scattering equationsfor the configurations |BGni and

B2

n

↵with the choice

of our Hamiltonian Eq. (73) are shown in Appendix HEqs. (H1) and (H2) respectively. We find that the scat-

Figure 1.9: S = 2 AKLT ground state with two dimers between nearest neighbors.Spin-S AKLT would have S dimers. |2G〉 with periodic boundary conditions.

αJ that satisfies the required condition. The Hamiltonian is then,

H(S) =∑i

(1 +

2S∑j=S

βj(c†i,i+1)

j(ci,i+1)j

)(1.37)

As we will show later in this section, this choice of coefficients is crucial for us

to have non-trivial exact states (including a tower of states) in the bulk of the

spectrum. Since the algebra of dimers described in Appendix A.1 is independent

of the spin model we are working with, it holds here as well.

1.6.1 Lower spectrum states

We start our analysis of the spin-S AKLT model with its ground state. Working

in the spin basis, if S dimers are formed between two spin-S (i and i + 1) that

have a total of 4S spin-1/2 Schwinger bosons, the maximum spin of both spin-S

combined cannot exceed S. Such a configuration must therefore be annihilated by

all P (J,S)i,i+1 for J > S, and is thus the unique ground state |SG〉 of the Hamiltonian

Eq. (1.32). This can be written as

|SG〉 =L∏i=1

(c†i,i+1)S | 0 〉 . (1.38)

The cartoon picture for the ground state of the spin-2 AKLT model is shown in

Fig. 1.9.

45

One might wonder if any low energy s = 0 magnons similar to the two Arovas

magnon states discussed in Sec. 1.4 exist for the spin-S AKLT models. We have

used the method described in Sec. 1.3 to detect exact eigenstates for S = 2 with

L ≤ 10 and S = 3 with L ≤ 8. In both the systems, we find only one singlet exact

state apart from the ground state, and it lies in the quantum number sector s = 0,

k = π, I = −1 for even L, and has an energy E = 2. Since the Arovas B state in

Sec. 1.4.2 is in the same sector, we need a similar configuration with short-ranged

dimers. Numerically, we do not find an analogue of the Arovas A state for higher

spin AKLT models with the chosen Hamiltonian given by Eq. (1.37) (i.e. with

αJ = 1 ∀J in Eq. (1.32)). At the end of this section, we provide an intuitive

explanation as to why this is the case. Note that while an Arovas A state cannot

be obtained for the Hamiltonian Eq. (1.37), an analogue can be constructed with

another suitable choice of coefficients αJ in Eq. (1.32) [55]. To exemplify the

derivation of the generalized Arovas B state, we first focus on the spin-2 AKLT

model. The corresponding Hamiltonian Eq. (1.37) reduces to

H(2) =∑i

(1− 1

84c†

2

i,i+1c2i,i+1 +

1

630c†

3

i,i+1c3i,i+1 −

1

6720c†

4

i,i+1c4i,i+1

). (1.39)

One could form spin-2 basis states by gluing together two copies of spin-1 basis

states, along with completely symmetrizing the spin-1s of the two copies. The

spin-2 configuration |BGn〉 (Fig. 1.10a) is formed by gluing one one spin-1 Arovas

B configuration |Bn〉 (Fig. 1.4a) to the spin-1 ground state |G〉 (Fig. 1.2) and

|BGn〉 (Fig. 1.10b) by gluing two |Bn〉s together. The scattering equations for

the configurations |BGn〉 and |B2n〉 with the choice of our Hamiltonian Eq. (1.37)

46

14c c c c c c c cc c c c c c c cc c c c c c c cc c c c c c c c- - - - - - - - -- - - - - - - - -

FIG. 14. S = 2 AKLT ground state with two dimers betweennearest neighbors. Spin-S AKLT would have S dimers. |2Giwith periodic boundary conditions.

Using Eq. (71) and Eq. (B7) for normal ordering, weobtain an expression for the spin-S AKLT Hamiltonian,

H(S) =X

i

0

@1 +2SX

j=1

j(c†i,i+1

)j(ci,i+1

)j

1

A (72)

where the coecients j depend on the coecients ↵J inthe projectors and the spin S, a closed form of which wecould not obtain for a general S. We have the freedom tochoose (S1) ↵J coecients while retaining the standardground state but changing the excitation spectrum. Bychoosing ↵J = 1 for all J , we observe that it is possible toset j = 0, 1 j S1. This is the only choice of ↵Jthat satisfies the required condition. The Hamiltonian isthen,

H(S) =X

i

0

@1 +2SX

j=S

j(c†i,i+1

)j(ci,i+1

)j

1

A (73)

As we will show later in this section, this choice of coef-ficients is crucial for us to have non-trivial exact states(including a tower of states) in the bulk of the spectrum.Since the algebra of dimers described in Appendix B isindependent of the spin model we are working with, itholds here as well.

A. Lower spectrum states

We start our analysis of the spin-S AKLT model withits ground state. Working in the spin basis, if S dimersare formed between two spin-S (i and i + 1) that havea total of 4S spin-1/2 Schwinger bosons, the maximumspin of both spin-S combined cannot exceed S. Such a

configuration must therefore be annihilated by all P (J,S)

i,i+1

for J > S, and is thus the unique ground state |SGi ofthe Hamiltonian Eq. (68). This can be written as

|SGi =LY

i=1

(c†i,i+1

)S |i . (74)

The cartoon picture for the ground state of the spin-2AKLT model is shown in Fig. 14. The ground state withopen boundary conditions is computed in Appendix J 4.

One might wonder if any low energy s = 0 magnonssimilar to the two Arovas magnon states discussed inSec. IV exist for the spin-S AKLT models. We have usedthe method described in Sec. III to detect exact eigen-states for S = 2 with L 10 and S = 3 with L 8.

c c c c c c cc c c c c c cc c c c c c cc c c c c c c- -- - - - - - - -- -- -

- -

nn-1 n+1 n+2n-2

(a)

c c c c c c cc c c c c c cc c c c c c cc c c c c c c- -- - - -- -

- -- -- -- -

nn-1n-2 n+1 n+2

(b)

c c c c c c cc c c c c c cc c c c c c cc c c c c c c-- - -- - - - --- --

---

nn-1 n+1 n+2n-2

(c)

c c c c c c cc c c c c c cc c c c c c cc c c c c c c- - -- - - --

--- -- --

-

nn-1n-2 n+1 n+2

(d)

c c c c c c cc c c c c c cc c c c c c cc c c c c c c- - - -- - - --- -

- ---

-nn-1 n+1 n+2n-2

(e)

FIG. 15. The spin-2 AKLT configurations (a) |BGni and(b)

B2n

↵. Such configurations form the generalization of the

Arovas B state in the spin-S AKLT model. (c) An example ofa symmetric configuration that appears only in CS

B2

. (d) Anexample of a symmetric configuration that appears in bothCSBG and CS

B2

. (e) An example of a non-symmetric scatteringconfiguration |i.

In both the systems, we find only one singlet exact stateapart from the ground state, and it lies in the quantumnumber sector s = 0, k = , I = 1 for even L, and hasan energy E = 2. Since the Arovas B state in Sec. IVBis in the same sector, we need a similar configurationwith short-ranged dimers. Numerically, we do not findan analogue of the Arovas A state for higher spin AKLTmodels with the chosen Hamiltonian given by Eq. (73)(i.e. with ↵J = 1 8J in Eq. (68)). At the end of this sec-tion, we provide an intuitive explanation as to why thisis the case. In Appendix H 2, we prove that for S = 2,an Arovas A state cannot be obtained for the Hamilto-nian Eq. (73) but an analogue can be constructed withanother suitable choice of coecients ↵J in Eq. (68).To exemplify the derivation of the generalized Arovas

B state, we first focus on the spin-2 AKLT model. Thecorresponding Hamiltonian Eq. (73) reduces to

H(2) =Pi

1 1

84

c†2

i,i+1

c2i,i+1

+ 1

630

c†3

i,i+1

c3i,i+1

1

6720

c†4

i,i+1

c4i,i+1

. (75)

One could form spin-2 basis states by gluing togethertwo copies of spin-1 basis states, along with completelysymmetrizing the spin-1s of the two copies. The spin-2configuration |BGni (Fig. 15a) is formed by gluing oneone spin-1 Arovas B configuration |Bni (Fig. 5a) to thespin-1 ground state |Gi (Fig. 2) and |BGni (Fig. 15b)by gluing two |Bnis together. The scattering equationsfor the configurations |BGni and

B2

n

↵with the choice

of our Hamiltonian Eq. (73) are shown in Appendix HEqs. (H1) and (H2) respectively. We find that the scat-

Figure 1.10: The spin-2 AKLT configurations (a) |BGn〉 and (b) |B2n〉. Such

configurations form the generalization of the Arovas B state in the spin-S AKLTmodel. (c) An example of a symmetric configuration that appears only in CSB2 .(d) An example of a symmetric configuration that appears in both CSBG and CSB2 .(e) An example of a non-symmetric scattering configuration |ζ〉.

assume the form

H(2) |BGn〉 = 2 |BGn〉+∑

η∈CSBG λ

BGη |η〉+ x

∑ζ∈CN λζ |ζ〉,

H(2) |B2n〉 = 2 |B2

n〉+∑

η∈CSB2

λB2

η |η〉 − 4∑

ζ∈CN λζT ±1 |ζ〉+ (1− x)∑

ζ∈CN λζ |ζ〉.(1.40)

In Eq. (1.40), T is a translation operator that translates by one site to the right

and λBGη , λB2

η and λζ are the scattering coefficients. We show an example of

a configuration from each of the sets CSB2 , CSBG and CN in Figs. 1.10c, 1.10d and

1.10e respectively. In Eq. (1.40), since non-vanishing projectors of the Hamiltonian

Eq. (1.37) act on on configurations |BGn〉 and |B2n〉 symmetrically about site n

(on bonds n − 2, n − 1 and n + 1, n + 2 in Figs. 1.10a and 1.10b), all the

47

scattering terms that are bond inversion symmetric (e.g. Fig. 1.10c) appear in

pairs that are related by a translation of an odd number of sites. Hence, they

cancel with a momentum π superposition of the configurations |BGn〉 and |B2n〉

for an even system size L. Moreover, |B2n〉 and |BGn〉 can be combined into

|2Bn〉 = 2 |BGn〉 −1

2

∣∣B2n

⟩, (1.41)

such that the scattering equation for |2Bn〉 reads

H(2) |2Bn〉 = 2 |2Bn〉+∑

η∈CS

λη |η〉+ 2∑

ζ∈CN

λζ(|ζ〉+ T ±1 |ζ〉

), (1.42)

where CS = CSBG∪CSB2 . In Eq. (1.42), the non-symmetric configurations also admit

a momentum π cancellation for L even. Thus, |2B〉 =∑

n (−1)n |2Bn〉 is an exact

state for L even with s = 0, k = π, and E = 2. In terms of spin operators,

using Eq. (1.35), we find that the normalized state ˜|2B〉 can be written as

˜|2B〉 = N∑n

(−1)n(−5 + Sn−1 · Sn + Sn · Sn+1 +

1

3Sn−1 · Sn, Sn · Sn+1

)2 ˜|2G〉,(1.43)

where , denotes the anti-commutator, ˜|2G〉 is the normalized ground state of

the spin-2 AKLT Hamiltonian and N a normalization factor. Note that the j = 1

term in general Hamiltonian Eq. (1.36) scatters the state |B2n〉 into non-symmetric

configurations (e.g. Fig. 1.10d) whereas it scatters |BGn〉 into symmetric config-

urations; this precludes the possibility of cancellation of non-symmetric terms as

earlier. We thus set β1 = 0, justifying our choice of the Hamiltonian in Eq. (1.37).

Moving on to the spin-S AKLT model, the set of configurations that can

48

be derived from the Arovas B state |Bn〉 and the spin-1 ground state |G〉 is

S = ∣∣BmGS−m⟩, 1 ≤ m ≤ S, which is obtained by gluing m spin-1 |Bn〉s

with S − m spin-1 |G〉s and completely symmetrizing the corresponding spins.

The derivation of the generalized exact states proceeds in a way similar to that of

S = 2. Eq. (1.37) scatters the configurations in the set S into two types of config-

urations: symmetric under bond inversion symmetry and non-symmetric. Similar

to the S = 2 case, we find that the non-symmetric scattering terms of∣∣BmGS−m⟩

are the same as the non-symmetric scattering terms of either of∣∣Bm+1GS−m−1

⟩or∣∣Bm−1GS−m+1

⟩. For S ≤ 5, we find that a |SBn〉 can always be constructed

from the configurations in the set S such that the scattering equation of |SBn〉

reads

H(S) |SBn〉 = 2 |SBn〉+∑η∈CS

λη |η〉+∑ζ∈CN

λζ(|ζ〉+ T 2xζ+1 |ζ〉), (1.44)

where CS and CN are the sets of bond inversion symmetric and non-symmetric

configurations of all the configurations in S and xζ is an integer. In Eq. (1.44),

the bond inversion symmetric configurations appear in pairs and vanish for even

L under a momentum-π superposition of |SBn〉. The non-symmetric terms too

appear in pairs, and hence vanish under the same conditions. We have analytically

derived the generalized Arovas B state up to S = 5. For generic S, we conjecture

the following expression for |SBn〉,

|SBn〉 =S∑

m=1

(−1)m

m

(S

m

) ∣∣BmGS−m⟩. (1.45)

This reduces to |Bn〉 and |2Bn〉 of Eq. (1.41) for S = 1 and S = 2 respectively.

49

The normalized exact state is then

|SB〉 = N∑n

(−1)n |SBn〉, (1.46)

where N is a normalization factor. This state has s = 0, k = π, E = 2. In spite

of the elegant form of Eq. (1.45) in terms of dimers, we could not easily find a nice

expression such as Eq. (1.43) for the state |SB〉 in terms of the spin operators.

As mentioned earlier, for our choice of the Hamiltonian Eq. (1.37), we do not

find an analogue of the Arovas A state numerically for S = 2 or S = 3. Analyti-

cally, an obstacle encountered is that some of the scattering terms of∣∣AmGS−m⟩

(the state obtained by gluing m spin-1 Arovas A configurations with S−m spin-1

ground states) are of the form of∣∣AnGS−n⟩. Such terms are bond inversion sym-

metric (for example, |A2〉 shown in Fig. 1.10d), thus precluding a cancellation with

momentum π (since the Arovas A configuration is also bond inversion symmetric).

Moreover, for any superposition of the configurations ∣∣AmGS−m⟩, the scattering

terms ∣∣AnGS−n⟩ appear in superposition with a different set of coefficients, thus

precluding the construction of an exact state. However, for S = 2, we find that a

fine-tuning of the Hamiltonian yields an Arovas A configuration with a scattering

equation similar to Eq. (1.44), and hence an Arovas A exact excited state.

1.6.2 Mid-spectrum states

In this section, we derive the generalization of the tower of states described in

Sec. 1.5.2 for the spin-S AKLT model. We start with a configuration gluing S

50

16

c c c c c c cc c c c c c cc c c c c c cc c c c c c c- - -- - - - - -- - -6 66 66 66 6nn-1 n+1

(a)

c c c c c c cc c c c c c cc c c c c c cc c c c c c c- - -- - - - -- ---6 66 66 66 6nn-1 n+1

(b)

FIG. 16. (a) Spin-4 Magnon |2Mni and (b) Scattering state|2Nni for spin-2 AKLT model. A similar picture holds forspin-S AKLT model.

thus precluding a cancellation with momentum (sincethe Arovas A configuration is also bond inversion sym-metric). Moreover, for any superposition of the configu-rations

AmGSm↵, the scattering terms

AnGSn↵

appear in superposition with a di↵erent set of coecients,thus precluding the construction of an exact state. How-ever, in Appendix H 2, we show that for S = 2, a fine-tuning of the Hamiltonian yields an Arovas A configura-tion with a scattering equation similar to Eq. (80), andhence an Arovas A exact excited state.

B. Mid-spectrum states

In this section, we derive the generalization of thetower of states described in Sec. VB for the spin-S AKLTmodel. We start with a configuration gluing S spin-1|Mni states to obtain a spin-2S magnon |SMni.

|SMni =Qn2

j=1

(c†j,j+1

)S(a†n1

)S(a†n)2S(a†n+1

)S

QLj=n+1

(c†j,j+1

)S |i . (83)

For example, |2Mni for S = 2 is shown in Fig. 16a. Withj = 0 for 0 < j < S, the Hamiltonian Eq. (73) does nothave any terms (c†)scs for s < S. Due to Eq. (G2), |2Mnivanishes under the action of (c†)mcm for m > S. Thus,using Eqs. (G3), the only term in the Hamiltonian thatcontributes to scattering is (c†)ScS . Thus, from Eq. (G3),the only scatterings term are those with S dimers on thebonds n 1, n or n, n+ 1, denoted by |SNn1

i and|SNni respectively, and shown for S = 2 in Fig. 16b.With this, we find the scattering equation of |SMni,

H(S) |SMni = 2 |SMni+ S(|SNn1

i+ |SNni). (84)

In the above equation, the precise value of the coecientS does not matter. For S = 2, S=2

= 2/7 but it ishard to obtain a closed form for in terms of S since itinvolves the normal ordering recursion relations Eq. (B8).From Eq. (84), the exact state is

|SS2

i =X

n

(1)n |SMni. (85)

In terms of spins, this can be expressed as

|SS2

i = NLX

n=1

(1)n(S+

n )2S]|SGi (86)

c c c c c c c cc c c c c c c cc c c c c c c cc c c c c c c c6 6 6 6 6 6 6 66 6 6 6 6 6 6 66 6 6 6 6 6 6 66 6 6 6 6 6 6 6(a)

c c c c c c c cc c c c c c c cc c c c c c c cc c c c c c c c-6 6 6 6 6 6 6 66 6 6 6 6 6 6 66 6 6 66 6 6 6 6 66 6 6 6

n n+1

(b)

c c c c c c c cc c c c c c c cc c c c c c c cc c c c c c c c6 6 6 6 6 66 6 6 6 6 66 6 6 6 6 66 6 6 6 6 6(c)

- -- -

c c c c c c c cc c c c c c c cc c c c c c c cc c c c c c c c-6 6 6 6 6 66 6 6 6 6 66 6 6 6 66 66 6 6

n n+1

(d)- -- -

FIG. 17. Upper spectrum state configurations for S = 2 (a)The ferromagnetic state with s = 2L and E = L. (b) Anexample of a non-scattering configuration with s = 2L 1,E = L. (c) A configuration that forms an exact state forS = 2 with E = L2, s = 2L4. (d) A dressed configurationthat forms an exact state with E = L 2, s = 2L 5.

where N is a normalization factor. Thus, we have anexact state for the spin-S AKLT model that closely re-sembles the spin-2 magnon of the spin-1 AKLT model.This state has E = 2, k = and s = 2S. Similar tothe spin-2 magnon of the spin-1 AKLT model, the spin-2S magnon generalizes for open boundary conditions aswell, see Appendix J 5.As for the spin-1 AKLT model tower of states, the

state with N spin-2S magnons on the ground state is alsoexact for the spin-S AKLT model. We denote the spin-2Smagnon creation operator as P =

PLn=1

(1)n(S+

n )2S .The tower of exact states is then

|SS2N i = PN |SGi . (87)

where 0 N L/2. |S2N i has k = 0 or depending on

whether N is odd or even, a total spin s = 2SN , andan energy of E = 2N . This tower of states connectsthe ground state to the ferromagnetic state. The proofproceeds exactly in the same way as for the spin-1 AKLTtower of states. As with the spin-1 AKLT tower of statesin Sec. VC, we conjecture that the tower of states for anyspin S lies in the bulk of the energy spectrum, althoughit is hard to obtain any strong numerical evidence of thisfor S > 1.

C. Upper spectrum states

In this section, we briefly comment on the structureof the simple upper spectrum excited states of the spin-S AKLT model. For the spin-S AKLT model, there isno unique highest excited multiplet. To see this, notethat apart from a constant, the Hamiltonian Eq. (73)contains terms (c†ij)

m(cij)m for m S. According toEq. (G2), all such terms vanish on any configuration withs < S dimers on one bond. For example, the configura-tion shown in Fig. 17b does not scatter and contributesone unit of energy, same as the fully ferromagnetic state

Figure 1.11: (a) Spin-4 Magnon |2Mn〉 and (b) Scattering state |2Nn〉 for spin-2AKLT model. A similar picture holds for spin-S AKLT model.

spin-1 |Mn〉 states to obtain a spin-2S magnon |SMn〉.

|SMn〉 =n−2∏j=1

(c†j,j+1)S(a†n−1)

S(a†n)2S(a†n+1)

S

L∏j=n+1

(c†j,j+1)S | 0 〉 . (1.47)

For example, |2Mn〉 for S = 2 is shown in Fig. 1.11a. With βj = 0 for 0 < j < S,

the Hamiltonian Eq. (1.37) does not have any terms (c†)scs for s < S. Due to

Eq. (A.14), |2Mn〉 vanishes under the action of (c†)mcm for m > S. Thus, using

Eqs. (A.15), the only term in the Hamiltonian that contributes to scattering is

(c†)ScS. Thus, from Eq. (A.15), the only scatterings term are those with S dimers

on the bonds n−1, n or n, n+1, denoted by |SNn−1〉 and |SNn〉 respectively,

and shown for S = 2 in Fig. 1.11b. With this, we find the scattering equation of

|SMn〉,

H(S) |SMn〉 = 2 |SMn〉+ λS(|SNn−1〉+ |SNn〉). (1.48)

In the above equation, the precise value of the coefficient λS does not matter. For

S = 2, λS=2 = −2/7 but it is hard to obtain a closed form for λ in terms of S since

it involves the normal ordering recursion relations Eq. (A.7). From Eq. (1.48), the

exact state is |SS2〉 =∑

n (−1)n |SMn〉. In terms of spins, this can be expressed

as

|SS2〉 = NL∑n=1

(−1)n(S+n )

2S |SG〉 (1.49)

51

where N is a normalization factor. Thus, we have an exact state for the spin-

S AKLT model that closely resembles the spin-2 magnon of the spin-1 AKLT

model. This state has E = 2, k = π and s = 2S. Similar to the spin-2 magnon

of the spin-1 AKLT model, the spin-2S magnon generalizes for open boundary

conditions as well [55].

As for the spin-1 AKLT model tower of states, the state with N spin-2S

magnons on the ground state is also exact for the spin-S AKLT model. We

denote the spin-2S magnon creation operator as P =∑L

n=1 (−1)n(S+n )

2S. The

tower of exact states is then

|SS2N〉 = PN |SG〉 . (1.50)

where 0 ≤ N ≤ L/2. |S2N〉 has k = 0 or π depending on whether N is odd or

even, a total spin s = 2SN , and an energy of E = 2N . This tower of states

connects the ground state to the ferromagnetic state. The proof proceeds exactly

in the same way as for the spin-1 AKLT tower of states. As with the spin-1 AKLT

tower of states in Sec. 1.5.3, we conjecture that the tower of states for any spin S

lies in the bulk of the energy spectrum, although it is hard to obtain any strong

numerical evidence of this for S > 1.

1.7 Conclusion

In this chapter, we have for the first time derived a tower exact eigenstates with

a closed-form expression in non-integrable models, the spin-S AKLT models. For

that purpose, we first used finite size exact diagonalizations to look for states with

a low rank of their reduced density matrix. These turned out to usually coincide

52

with a rational or even an integer energy. For each of them, we have then derived

an analytical formula. Apart from the tower of states from the ground state to

the highest excited state, we have obtained several exact excited states in the low

energy and the high energy spectrum. Our approach could potentially be applied

to any non-integrable model, irrespective of its dimensionality.

In the context of the Eigenstate Thermalization Hypothesis, some of the exact

states we have obtained seem to be located in the bulk of the spectrum but still

have non-generic entanglement properties, which questions the strong ETH. As

we discuss in Chapter 2 those states have a low entanglement entropy, and it

is unlikely that any of them would be thermal. Our results pave the way for

the search of non-integrable models that provide some analytical insight on the

Eigenstate Thermalization Hypothesis. They also suggest that a special class of

“semi-solvable” but non-integrable spin models exist [127,136,137]. These models

would contain thermal and non-thermal eigenstates mixed together [53, 68, 138],

something that is usually thought to be impossible in non-integrable models.

53

L = 14E D k s I Pz |ψ〉0 1 0 0 1 1 |G〉53

1 π 0 1 1 |A〉2 1 π 0 -1 1 |B〉2 1 π 2 -1 1 |S2〉4 1 0 4 1 1 |S4〉6 1 π 6 -1 1 |S6〉8 1 0 8 1 1 |S8〉10 1 π 10 -1 1 |S10〉12 5 π 12 (−1)n 1 |2n〉, n = 1, 3− 612 1 2nπ

712 - 1 |2k〉

12 1 π 13 1 -1 |1k=π〉383

1 π 12 1 1 |2n=0〉14 1 0 14 1 1 |F 〉

L = 16E D k s I Pz |ψ〉0 1 0 0 1 1 |G〉53

1 π 0 1 1 |A〉2 1 π 0 -1 1 |B〉2 1 π 2 -1 1 |S2〉4 1 0 4 1 1 |S4〉6 1 π 6 -1 1 |S6〉8 1 0 8 1 1 |S8〉10 1 π 10 -1 1 |S10〉12 1 π 12 -1 1 |S12〉14 7 π 14 (−1)n 1 |2n〉, n = 1, 3− 814 1 π 15 1 -1 |1k=π〉443

1 π 14 1 1 |2n=0〉16 1 0 16 1 1 |F 〉

Table 1.1: Table showing the highest weight states with rational energies forL = 14 and L = 16 (sectors k = 0, π). E is energy, D is the degeneracy (excludingthe SU(2) multiplet degeneracy), k momentum, s the spin quantum number, Ithe eigenvalue under bond inversion symmetry, Pz is the eigenvalue under spinflips for the Sz = 0 state of the multiplet and |ψ〉 the state we identify it with.The states in this table with s < L− 2 have a very sparse entanglement spectrumcompared to typical states in their quantum number sectors.

54

33

P (2,1)ij

Es sc c c cc c c cm m m m- - -

m i j n

= 0 (D6)

P (2,1)ij

Es s s sc c c c c cc c c c c cm m m m m m- -

- -m n ji p r

=

Es s s sc c c c c cc c c c c cm m m m m m- -- -

m n ji p r

+14

Es s s sc c c c c cc c c c c cm m m m m m- --

-m n ji p r

+

Es s s sc c c c c cc c c c c cm m m m m m--

--

m n ji p r

+

Es s s sc c c c c cc c c c c cm m m m m m---

-m n ji p r

+

Es s s sc c c c c cc c c c c cm m m m m m- ---

m n ji p r

+112

Es s s sc c c c c cc c c c c cm m m m m m-

--

-m n ji p r

+

Es s s sc c c c c cc c c c c cm m m m m m--- -

m n ji p r

(D7)

P (2,1)ij

Ec cc cm m6 66 6

ji

=

Ec cc cm m6 66 6ji

(D22)

P (2,1)ij

Esc c cc c cm m m66 6-

jin

=

Esc c cc c cm m m66 6-jin

+12

Esc c cc c cm m m66 6-

jin

(D23)

P (2,1)ij

Es sc c c cc c c cm m m m- -6 6

m i j n

=

Es sc c c cc c c cm m m m- -6 6m i j n

+112

Es sc c c cc c c cm m m m-6 6

-

m i j n

+14

Es sc c c cc c c cm m m m- - 6 6

m i j n

+

Es sc c c cc c c cm m m m- -6 6m i j n

+

Es sc c c cc c c cm m m m-6 6-

m i j n

(D24)

P (2,1)ij

Es sc c c cc c c cm m m m-

-66ji p r

=

Es sc c c cc c c cm m m m--66

ji p r

+16

Es sc c c cc c c cm m m m- 6 6

-

ji p r

+12

Es sc c c cc c c cm m m m- -6 6

ji p r

+

Es sc c c cc c c cm m m m--6 6

ji p r

(D25)

P (2,1)ij

Es s sc c c c cc c c c cm m m m m-

- -6m ji p r

=

Es s sc c c c cc c c c cm m m m m-- -6

m ji p r

+14

Es s sc c c c cc c c c cm m m m m- - - 6

m ji p r

+

Es s sc c c c cc c c c cm m m m m-- - 6

m ji p r

+

Es s sc c c c cc c c c cm m m m m- -6-

m ji p r

+

Es s sc c c c cc c c c cm m m m m--6-

m ji p r

+124

Es s sc c c c cc c c c cm m m m m- 6

--

m ji p r

+

Es s sc c c c cc c c c cm m m m m- 6--

m ji p r

(D26)

P (2,1)ij

Ec cc cm m6 6-

ji

= 0 (D27)

P (2,1)ij

Esc c cc c cm m m6- -

jim

= 0 (D28)

FIG. 27. Diagrammatic representation of the action of the projector P (2,1)ij

on various configurations of dimers around bondi, j. The configurations of the filled small circles are not relevant to the scattering equation and are the same (in terms of theSchwinger bosons) on both sides of a given equation. The directions of the arrows are crucial; reversing an arrow contributesa factor of (1).

Figure 1.12: Diagrammatic representation of the action of the projector P (2,1)ij on

various configurations of dimers around bond i, j. The configurations of thefilled small circles are not relevant to the scattering equation and are the same onboth sides of a given equation. Reversing the direction of an arrow contributes afactor of (−1). 55

2Entanglement of Quasiparticle Excited

States

2.1 Introduction

The entanglement structure of low-energy excitations in integrable and non-integrable

models has been studied analytically and numerically in detail [139–146], partic-

ularly using the language of Matrix Product States (MPS) [108, 110]. Similar

56

to the ground states of gapped Hamiltonians [112], low-energy excited states of

gapped Hamiltonians are in principle also captured by this MPS framework [139].

However, even within single-mode excitations, the lack of explicit examples has

hindered a study of their entanglement in more detail; for example the general

nature of finite-size corrections to the entanglement spectra is unknown. Be-

yond low-energy excitations, the structure of excited states has been studied in

the MBL regime, where all the eigenstates exhibit area-law entanglement [33],

and consequently have an efficient MPS representation [147–149]. In the ther-

mal regime, however, very little is analytically known about the kind of excited

states that can exist in the bulk of the spectrum of generic non-integrable mod-

els [53,67,138,150–152]. For example, can certain highly excited states of thermal

non-integrable models have an exact or approximate matrix product structure

with a finite or low bond dimension in the thermodynamic limit?

In Chapter 1, we obtained a tower of exact excited states analytically in a

family of non-integrable models, the spin-S AKLT models. Being the first few

known examples of exact eigenstates of non-integrable models, we propose to use

the excited states of these models to test conjectures on eigenstates that exist

in the literature. We recover the general entanglement spectra of single-mode

excitations, earlier obtained on general grounds [139, 140]. We also derive the

entanglement spectrum of an entire tower of exact states, thus generalizing the

single-mode results to these set of states. The tower of states have an interesting

entanglement structure in that the zero energy density states entanglement spectra

is composed of shifted copies of the ground state entanglement spectrum. This

structure generalizes the earlier result obtained on the entanglement spectra of

SMA excitations. We find that the finite energy density states in the tower have a

57

sub-thermal entanglement entropy scaling in spite of the fact that they appear to

be in the bulk of the spectrum [55]. More precisely, the entanglement entropy S

for these states scales as S ∝ logL where L is the subsystem size. This indicates

a violation of the strong ETH [96, 97], which states that all the eigenstates in

the bulk of the spectrum of a non-integrable model in a given quantum number

sector are thermal, i.e. their entanglement entropy scales with the volume of the

subsystem (S ∝ L).

This chapter is organized as follows. In Sec. 2.2, we provide some examples

of Matrix Product States (MPS) and Matrix Product Operators (MPO) in the

context of the one-dimensional AKLT models. We will be using these tools to

study the entanglement spectra of the excited states. In Sec. 2.3, we discuss the

structure and properties of states that are created by the action of an operator

(MPO) on the ground state (MPS). In Sec. 2.4, we derive the entanglement spec-

tra of single-mode excitations, focusing on the AKLT Arovas states and spin-2S

magnons. In Secs. 2.5 and 2.6, we consider states beyond single-mode excita-

tions. We compute the entanglement spectrum of the tower of states in spin-S

AKLT models, where we work in the zero energy density and finite energy density

regimes separately. Further, in Sec. 2.7, we discuss the violation of the Eigenstate

Thermalization Hypothesis and then show numerical results away from the AKLT

point. In Sec. 2.8, we review symmetries and their effects on the entanglement

spectra of the ground states, and discuss symmetry-protected exact degeneracies

and finite-size effects in the entanglement spectra of the excited states.

58

c c c c c c c cc c c c c c c cm m m m m m m m- - - - - - -6 6

Figure 2.1: Ground State of the spin-1 AKLT model with Open Boundary Con-ditions. Big and small circles represent physical spin-1 and spin-1/2 Schwingerbosons respectively. The lines repesent singlets between spin-1/2. The two edgespin-1/2’s are free.

2.2 MPS and MPO in the AKLT models

2.2.1 MPS

In this section, we provide a few examples of MPS and MPO based on the AKLT

models.

Ground state of the spin-1 AKLT model

We first focus on the ground state of the spin-1 AKLT model with Open Boundary

Conditions (OBC) [54], one of the first examples of an MPS [153]. As discussed in

Chapter 1, the state with L spin-1’s can be thought to be composed of two spin-

1/2 Schwinger bosons, each in a singlet configuration with the spin-1/2 Schwinger

boson of the left and right nearest neighbor spin-1s. Thus there are dangling

spin-1/2’s on each edge of the chain. A cartoon picture of this state is shown in

Fig. 2.1.

The two spin-1/2 Schwinger bosons within a spin-1 (see Fig. 2.1) form a virtual

Hilbert space that corresponds to the auxillary space of the MPS. The normalized

wavefunction can be written as a matrix product state with physical dimension

d = 3 (the Hilbert space dimension of the physical spin-1) and a bond dimension

χ = 2 (the Hilbert space dimension of the spin-1/2 Schwinger boson). The d

59

normalized χ× χ matrices for the AKLT ground state read [108]

A[1] =

√2

3

0 1

0 0

, A[0] =1√3

−1 0

0 1

, A[-1] =

√2

3

0 0

−1 0

(2.1)

corresponding to Sz = 1, 0, -1 of the physical spin-1 respectively. Using the ma-

trices of Eq. (2.1), the AKLT ground state transfer matrix can be computed to

be

E =

13

0 0 23

0 −13

0 0

0 0 −13

0

23

0 0 13

(2.2)

where the left and right indices of the transfer matrix are grouped together. The

eigenvalues of this transfer matrix are (1,−13,−1

3,−1

3). Since the largest eigenvalue

is non-degenerate, using Eq. (25) the AKLT groundstate is a finitely-correlated

state with correlation length ξ = 1/ log(3). The boundary vectors of Eq. (18) for

the AKLT ground state correspond to the free spin-1/2’s on the left and right

edges of an open spin-1 chain, shown in Fig. 2.1. With both edge spins set to

Sz = +1/2 the boundary vectors are

blA =

1

0

brA =

0

1

. (2.3)

The Gram matrices L and R for the AKLT ground state are the left and right

eigenvectors of E corresponding to eigenvalue 1, L = R = 1√212×2. Using Eq. (28)

the reduced density matrix is ρred = 1212×2 and the entanglement entropy is S =

60

c c c c c c c cc c c c c c c cc c c c c c c cc c c c c c c c - - - - - - -6 66 6- - - - - - -

Figure 2.2: Spin-2 AKLT model ground state with 2 singlets between nearestneighbors. The four edge spin-1/2s are free. Spin-S AKLT has S singlets.

log 2, corresponding to a free spin-1/2 dangling spin.

Ground state of the spin-S AKLT model

In a spin-S chain, each of the physical spin-S can be thought of as composed of 2S

spin-1/2 Schwinger bosons, or equivalently, two spin-(S/2) bosons. The ground

state of the spin-S AKLT model then has S singlets between the 2S Schwinger

bosons (S on each site) on neighboring sites, as shown for S = 2 in Fig. 2.2. It can

also be interpreted as having a “spin-(S/2) singlet” between the spin-(S/2)’s of

neighboring sites. Here, a spin-(S/2) singlet is the state formed by two spin-(S/2)

with a total spin J = 0, Jz = 0. In the case of S = 1, this coincides with a usual

spin-1/2 singlet. Consequently, with OBC, there are two free spin-(S/2)’s that

set the boundary conditions of the wavefunction (see Fig. 2.2).

An MPS representation for the spin-S AKLT ground state can be developed in

close analogy to the spin-1 AKLT ground state [153–158]. Here as well, the virtual

Hilbert space of the spin-S/2 bosons corresponds to the auxiliary space. Thus,

the MPS physical dimension is d = 2S + 1 (because of spin-S physical spins) and

the bond dimension is χ = S + 1 (because of the spin-S/2 virtual spins). The

χ× χ MPS matrices of the spin-S AKLT ground states have the form

A[m]αβ = κmαβδα−β,m (2.4)

where κmαβ is a non-vanishing constant [56] that is not important for the results

61

of this chapter.

Analogous to Eq. (2.3), the boundary vectors of the MPS corresponding to

boundary conditions with both the edge spin-(S/2)’s with Sz = +S/2 are χ-

dimensional vectors with components

(blA)α = δα,1 (brA)α = δα,χ. (2.5)

Indeed one can verify that the spin-S AKLT ground state of Eq. (2.4) is finitely

correlated, and the left and right eigenvectors corresponding to the largest eigen-

value 1 are both L = R = 1χ×χ. Thus the reduced density matrix reads

ρred =1

S + 11(S+1)×(S+1) (2.6)

and the entanglement entropy is S = log(S + 1).

2.2.2 MPO

We now introduce the MPOs for some of the operators required to build ex-

act excited states of the AKLT model. These will be useful for the study of

the entanglement of these excited states. Whereas the Arovas A and Arovas B

states discussed in Chapter 1 were for exact eigenstates only for periodic bound-

ary conditions, here we assume open boundary conditions. The motivation for

this assumption is twofold. First, analytic calculations using MPS and MPOs are

greatly simplified with open boundaries. Second, we are interested in the thermo-

dynamic limit or large systems where the properties of the system are essentially

independent of boundary conditions.

62

We start with the spin-1 AKLT model. The closed-form expression for the

Arovas A state up to an overall normalization factor for OBC reads

|A〉 =

[L−1∑j=1

(−1)jSj · Sj+1

]|G〉 (2.7)

where |G〉 is the ground state of the spin-1 AKLT model and we have assumed

open boundary conditions. The operator that appears in the Arovas A state can

be written as

OA =∑j

(−1)jSj · Sj+1 =∑j

(−1)j(S+j S

−j+1 + S−

j S+j+1

2+ SzjS

zj+1

). (2.8)

By analogy to the MPO of Eq. (B.12) corresponding to the operator Eq. (B.11),

the MPO for OA (in the case of open boundary conditions) reads

MA =

−1 −S+√2−S−

√2−Sz 0

0 0 0 0 S−√2

0 0 0 0 S+√2

0 0 0 0 Sz

0 0 0 0 1

, (2.9)

where the negative signs appear due to the (−1)j in Eq. (2.8). Similarly, the

expression for the Arovas B state with OBC can be written as |B〉 = OB |G〉

where

OB =L−1∑j=2

(−1)jSj−1 · Sj, Sj · Sj+1. (2.10)

The MPO for OB can be compactly expressed as

63

MB =

−1 −S 0 0

0 0 T 0

0 0 0 S

0 0 0 1

,

S =

(S+√2

S−√2

Sz), S =

(S−√2

S+√2

Sz)T

T =

S−,S+

2(S−)2 S−,Sz√

2

(S+)2 S+,S−2

S+,Sz√2

Sz ,S+√2

Sz ,S−√2

2SzSz

. (2.11)

The bond dimension of the MPO MB is thus χm = 8.

Another set of excited states for spin-S AKLT models were the spin-2S magnons.

The closed-form expression for the spin-2S magnon state in the spin-S AKLT

models, up to an overall normalization factor, reads

|SS2〉 =L∑j=1

(−1)j(S+j )

2S |SG〉 , (2.12)

where |SG〉 is the ground state of the spin-S AKLT model. Unlike the two previous

states, |SS2〉 is an exact excited state irrespective of the boundary conditions.

The spin-2S magnon creation operator and its bond dimension χm = 2 MPO (by

analogy to Eqs. (B.5) and (B.9)) thus read

OSS2 =∑j

(−1)j(S+j )

2S, MSS2 =

−1 −(S+)2S

0 1

(2.13)

As discussed in Chapter 1, the spin-S AKLT models consisted of a tower of

exact eigenstates from the ground state to a highest excited state comprised of

multiple spin-2S magnons. The closed-form expression for the N -th state of the

64

tower of states for the spin-S AKLT model reads

|SS2N〉 = (OSS2)N |SG〉 . (2.14)

When written naively, the MPO for the operator (OSS2)N has a bond dimension

2N , since it is a direct product of N copies of the MPO MSS2 on the auxiliary

space. However, a more efficient MPO can be constructed for (OSS2)N . For

example, consider N = 2. (OSS2)2 can be written as (up to an overall factor)

(OSS2)2 =

∑i≤j

(−1)i+j(S+i )

2S(S+j )

2S =∑i

(−1)i(S+i )

2S∑i<j

(−1)j(S+j )

2S, (2.15)

where we have used that (S+j )

4S = 0. From Eq. (2.15) it is evident that the MPO

MSS4 for (OSS2)2 can be viewed as two copies of the generalized FSA generating

MSS2 , where the final state of the first generalized FSA is the initial state for the

second generalized FSA. The MPO thus reads

MSS4 =

−1 −(S+)2S 0

0 1 (S+)2S

0 0 −1

(2.16)

The appearance of three ±1 on the diagonal of MSS4 reflects the non-locality of

the operator (OSS2)2. The same strategy can be applied to construct the MPO

65

MSS2Ncorresponding to the operator (OSS2)

N . For general N , the MPO reads

MSS2N=

−1 −(S+)2S 0 . . . 0

0 1 (S+)2S. . . ...

... . . . . . . . . . 0

... . . . . . . (−1)N1 (−1)N(S+)2S

0 . . . . . . 0 (−1)N+11

. (2.17)

The bond dimension of the MPO MSS2Nis thus χm = N + 1.

2.3 MPO × MPS

As discussed in Chapter 1, the exact states of the spin-S AKLT models are ob-

tained by acting local operators on the ground states [55]. In this section we

study some of the properties of an MPS formed by acting an MPO (operator) on

an MPS with a finite correlation length (ground state). Similar approaches (e.g.

tangent space methods) have been used to study low energy excitations of gapped

Hamiltonians [139, 141, 142, 144, 159, 160].

2.3.1 Definition and properties

A state defined by the action of an MPO on an MPS (we assume both to be

site-independent) has a natural MPS description,

B[m] =∑n

M [mn] ⊗ A[n]. (2.18)

66

where the tensor product ⊗ acts on the ancilla. We refer to B as an MPO×MPS

to distinguish it from the MPS A, which we assume to have a finite correlation

length. B has a bond dimension of Υ = χmχ, where χm and χ are the bond

dimensions of the MPO and MPS respectively. Note that Υ need not be the

optimum bond dimension of B (i.e. Schmidt rank of the state B represents),

though it is typically the case when M and A have optimum bond dimensions.

The transfer matrix of B reads

F =∑m

B[m]∗ ⊗B[m] =∑m,n,l

A[m]∗ ⊗M [nm]∗ ⊗M [nl] ⊗ A[l] ≡∑m,l

A[m]∗ ⊗M[ml] ⊗ A[l],

(2.19)

where ⊗ acts on the ancilla and

M[ml] ≡∑n

M [nm]∗ ⊗M [nl] =∑n

M †[mn] ⊗M [nl], (2.20)

where † acts on the physical indices on the MPO. F is thus a Υ×Υ×Υ×Υ tensor

that can also be viewed as Υ2 × Υ2 matrix by grouping both the left and right

ancilla. From Eqs. (2.18) and (2.19), the boundary vectors of an MPO×MPS and

its transfer matrix are given by

blB = blM ⊗ blA, brB = brM ⊗ brA, blF = (bl∗B ⊗ blB) brF = (br∗B ⊗ brB). (2.21)

Since M is always upper triangular in the auxiliary indices (as discussed in

Sec. B.1), M is a χ2m × χ2

m matrix with a nested upper triangular structure in

the ancilla, with elements as d× d matrices, where d is the physical Hilbert space

67

dimension. For example, if we consider the MPO of Eq. (B.9), M reads

M =

1 C C† C†C

0 e−ik1 0 e−ikC†

0 0 eik1 eikC

0 0 0 1

. (2.22)

In Eq. (2.19), the matrix elements of F can also be viewed as a χ2m × χ2

m matrix

with matrix elements

Fµν =∑m,l

A[m]∗ ⊗M[ml]µν A

[l]. (2.23)

Fµν is indeed the generalized transfer matrix (see Eq. (22)) of the operatorMµν .

Thus, F is also a nested upper triangular matrix with elements χ2×χ2 generalized

transfer matrices of the elements of M with the original MPS A. For M of

Eq. (2.22), we obtain

F =

E EC EC† EC†C

0 e−ikE 0 e−ikEC†

0 0 eikE eikEC

0 0 0 E

(2.24)

where E is the transfer matrix of the MPS A and EC , EC† and EC†C are the

generalized transfer matrices (defined in Eq. (22)) of operators C, C† and C†C

respectively. Furthermore, since the MPO boundary conditions are always of the

form of Eq. (B.13), using Eq. (2.21) the boundary vectors for the transfer matrix

68

F read

brF =

0

0

0

brE

blF =

blE

0

0

0

. (2.25)

Since non-vanishing diagonal elements of the MPO can only be of the form eiθ1.

Consequently, the diagonal elements of F are always of the form eiθE, as can

be observed in the example in Eq. (2.24). Note that the block upper triangular

structure of F dictates that its generalized eigenvalues are those of eiθE blocks

on the diagonal [56]. The eigenvalue of unit magnitude of the transfer matrix

F is thus not unique in general, and an MPO × MPS typically does not have

exponentially decaying correlations even if the MPS has.

Moreover, the transfer matrix F need not be diagonalizable. In general, it

has a Jordan normal form consisting of Jordan blocks corresponding to various

degenerate generalized eigenvalues. The Jordan decomposition of F reads

F = PJP -1, J =⊕i∈Λ

Ji, (2.26)

where J is the Jordan normal form of F , the columns of P are the right generalized

eigenvectors of F and the rows of P -1 are the left generalized eigenvectors of F

(same as right generalized eigenvectors of F T ). Furthermore, Λ is a set of indices

that label the Jordan blocks, Ji is a Jordan block of size |Ji| of an eigenvalue λi

69

and∑

i∈Λ |Ji| = Υ2. That is, up to a shuffling of rows and columns,

Ji =

λi 1 0 . . . . . . 0

0 λi 1. . . . . . ...

... . . . . . . . . . . . . ...

... . . . . . . . . . . . . 0

... . . . . . . . . . λi 1

0 . . . . . . . . . 0 λi

|Ji|×|Ji|

(2.27)

For a diagonalizable matrix, |Ji| = 1 for all i ∈ Λ.

2.3.2 Entanglement spectrum from an MPO × MPS

In this section, we outline the computation of the entanglement spectrum for an

MPO × MPS state, i.e., for an MPS with a non-diagonalizable transfer matrix.

Since the MPO ×MPS is also an MPS, Eqs. (26) to (28) of Sec. 0.2.1 in Chapter 0

are valid here as well. Analogous to Eq. (27), here we obtain

L = (F T )LAblF R = FLBbrF . (2.28)

In the following, we will mostly be interested in the limit n ≡ LA = LB → ∞,

i.e. the thermodynamic limit with an equal bipartition. Since F n = PJnP -1,

70

Jn =⊕

i∈Λ Jni , and

Jni =

λni(n1

)λn−1i

(n2

)λn−2i . . .

(n

|Ji|−1

)λn−|Ji|+1i

0 λni(n1

)λn−1i

. . . ...... . . . . . . . . .

(n2

)λn−2i

... . . . . . . λni(n1

)λn−1i

0 . . . . . . 0 λni

|Ji|×|Ji|

, (2.29)

all the Jordan blocks Ji corresponding to |λi| < 1, vanish in the thermodynamic

(n→∞) limit. We can thus truncate J to a subspace with generalized eigenvalues

of magnitude 1, by including a projector Q onto that subspace. This subspace

could involve several Jordan blocks, each of possibly different dimension. We

define

Junit = QJQ =⊕i∈Λunit

Ji, (2.30)

where Λunit is a set defined such that |λi| = 1 for i ∈ Λunit, and the dimension

of Junit is |Junit|, where |Junit| =∑

i∈Λunit|Ji|. Since we are interested in the

limit n → ∞, instead of F , we use a truncated transfer matrix Funit defined

as Funit ≡ PJunitP-1, such that F n

unit = F n as n → ∞ Since Q2 = Q, using

Eq. (2.30), the expression for Funit can be written as

Funit = PQ(QJQ)QP -1 ≡ VRJunitVTL , VR ≡ PQ, V T

L ≡ QP -1, (2.31)

where we have used Eq. (2.30). Since VR consists of the columns of P (right

generalized eigenvectors of F ) corresponding to the generalized eigenvalues in J

and V TL consists of the rows of P -1 (left generalized eigenvectors of F ), VR and VL

71

have the forms

VR =

(r1 r2 . . . r|Junit|

), VL =

(l1 l2 . . . l|Junit|

), (2.32)

where ri (resp. li) are the Υ2-dimensional right (resp. left) generalized eigen-

vectors of F corresponding the generalized eigenvalues of magnitude 1.

Using Eq. (2.31), the truncated Gram matrices read

Runit = VR(Junit)nV T

L brF , Lunit = VL(J

Tunit)

nV TR b

lF . (2.33)

We split Eq. (2.33) into two parts. We first define the |Junit|-dimensional “modi-

fied” boundary vectors that are independent of n as

βrF ≡ V TL b

rF , βlF ≡ V T

R blF . (2.34)

The n-dependent parts of Lunit and Runit are then encoded in the (Υ)2 × |Junit|

dimensional matrices

WR ≡ VR(Junit)n, WL ≡ VL(J

Tunit)

n. (2.35)

Since L and R are viewed as Υ × Υ matrices in Eq. (28), it is natural to view

the columns of L and R as Υ × Υ matrices in Eq. (2.35). Consequently, we can

directly view the columns of VL and VR (defined in Eq. (2.32)) as Υ×Υ matrices.

To obtain a direct relation between the generalized eigenvectors of F and the

projected Gram matrices Lunit and Runit (defined in Eq. (2.33)), we need to de-

termine how WL and WR depend on the generalized eigenvectors. Suppose the

72

components of WR and WL have the following forms

WR ≡(R1 R2 . . . R|Junit|

), WL ≡

(L1 L2 . . . L|Junit|

), (2.36)

where Ri and Li are Υ × Υ matrices. Runit and Lunit are n-independent

superpositions of the matrices Ri and Li. Their expressions read

Runit =

|Junit|∑i=1

Ri(βrF )i, Lunit =

|Junit|∑i=1

Li(βlF )i. (2.37)

To relate Ri and Li to ri and li, we need to consider the Jordan block

structure of Junit. If Junit consists of a single Jordan block of generalized eigenvalue

λ, dimension |Junit|, and of the form of Eq. (2.27); using Eqs. (2.29) and (2.35),

we directly obtain

Ri =i−1∑j=0

(n

j

)ri−jλ

n−j, Li =

|Junit|−i∑j=0

(n

j

)li+jλ

n−j, (2.38)

where ri and li are viewed as Υ×Υ matrices.

For Junit composed of several Jordan blocks, Ji, (e.g. in Eq. (2.30)), Eq. (2.38)

holds for each Jordan block separately. We first consider a subset of right and

left generalized eigenvectors of Funit, r(Jk)i ⊂ ri and l(Jk)i ⊂ li that are

associated with the Jordan block Jk of dimension |Jk| and generalized eigenvalue

λk, |λk| = 1. Here, we assume that r(Jk)1 (resp. l(Jk)1 ) is the right (resp. left)

eigenvector and r(Jk)i (resp. l

(Jk)i ) is the (i − 1)-th right (resp. left) generalized

eigenvector. We then define R(Jk)i ⊂ Ri and L(Jk)

i ⊂ Li that are related

73

to r(Jk) and l(Jk) as

R(Jk)i =

i−1∑j=0

(n

j

)r(Jk)i−j λ

n−jk , L

(Jk)i =

|Jk|−i∑j=0

(n

j

)l(Jk)i+j λ

n−jk . (2.39)

This is the analogue of Eq. (2.38) for a single Jordan block Jk. Using Eqs. (2.37)

and (2.39), Runit and Lunit are of the form

Runit =

|Junit|∑i=1

fR(i, n, βrF )ri, Lunit =

|Junit|∑i=1

fL(i, n, βlF )li, (2.40)

where fR(i, n, βrF ) and fL(i, n, βlF ) are scalar coefficients that depend on n

through Eq. (2.39) and on the boundary condition dependent vectors βrF and βlF

respectively.

Since Lunit andRunit are the same as L andR in the thermodynamic limit, using

Eq. (2.40), the unnormalized and usually non-Hermitian matrix ρred of Eq. (28)

that has the same spectrum as the reduced density matrix reads

ρred =

|Junit|∑i,j=1

fL(i, n, βlF )fR(j, n, β

rF )lir

Tj . (2.41)

In the limit of large n, ρred can be computed using Eq. (2.41) order by order in

n. Such a calculation will be discussed with concrete examples from the AKLT

models in the next three sections.

2.4 Single-mode Excitations

To illustrate the results of the previous section, we first consider single-mode ex-

citations. A single-mode excitation is defined as an excited eigenstate created by

74

a local operator acting on the ground state. It is known that such wavefunctions

are efficient variational ansatzes for low energy excitations of gapped Hamiltoni-

ans [139]. Such excitations, dubbed as Single-Mode Approximation (SMA) or the

Feynman-Bijl ansatz, have also been used as trial wavefunctions for low energy

excitations in a variety of models [122, 124, 139, 141, 161–163].

2.4.1 Structure of the transfer matrix

The SMA state obtained by a local operator O can be written as

∣∣Ok

⟩=∑j

eikjOj |G〉 ≡ Ok |G〉 . (2.42)

where Oj denotes the operator O in the vicinity of site j of the spin chain (if not

purely onsite), |G〉 is the ground state of the system and k is the momentum of

the SMA state. In the spin-1 AKLT model, the three low-lying exact states have

the form of Eq. (2.42) with k = π, i.e., the SMA generates an exact eigenstate.

In the language of matrix product states, SMA states can be represented as an

MPO×MPS, where the MPO represents the operator Ok, and the MPS is the

matrix product representation of the ground state |G〉. As discussed in Sec. B.1

in Appendix B, the MPO of a translation invariant local operator Ok defined

in Eq. (2.42) can be constructed such that it is upper triangular with only two

non-vanishing diagonal elements, eik1 and 1. This structure can also be observed

in the MPOs of the creation operators of the excited states of the AKLT model,

shown in Eqs. (2.9), (2.11) and (2.13). For the single-mode approximation, the

transfer matrix F of∣∣Ok

⟩thus has four non-vanishing blocks on the diagonal and

its generalized eigenvalues are those of the submatrices on the diagonal. Since all

75

the SMA states of the AKLT model are at momentum π, we set k = π in the

following. The same analysis holds for any k 6= 0.

We illustrate the entanglement spectrum calculation for the simplest case, where

F has the form of Eq. (2.24), corresponding to an MPO with bond dimension

χm = 2, the one in Eq. (B.9) and k = π,

F =

E EC EC† EC†C

0 −E 0 −EC†

0 0 −E −EC

0 0 0 E

, blF =

blE

0

0

0

brF =

0

0

0

brE

. (2.43)

2.4.2 Derivation of ρred

The generalized eigenvalues of F of Eq. (2.43) with a unit magnitude are +1,−1,−1,+1,

the largest eigenvalues of the submatrices E (the transfer matrices of the ground

state MPS). The +1 generalized eigenvalues in F generically form a Jordan block

for a typical operator Ok [56]. Since the off-diagonal block between the subspaces

of the two −E blocks is 0 (as seen in Eq. (2.43)), the two −1 generalized eigenval-

ues in F do not form a Jordan block. Thus, for a typical operator Ok, the Jordan

normal form Junit of the truncated transfer matrix Funit (defined in Eq. (2.31))

can be decomposed into three Jordan blocks as

Junit = J0 ⊕ J-1 ⊕ J1, J0 =

1 1

0 1

J-1 = (−1) J1 = (−1). (2.44)

76

Following the convention of Eq. (2.32), we assume that VR and VL have the forms

VR = (r1 r2 r3 r4), VL = (l1 l2 l3 l4) (2.45)

Since the +1 generalized eigenvalues are due to the top and bottom blocks of F ,

r1 (resp. l1) and r4 (resp. l4) are the right (resp. left) generalized eigenvectors

corresponding to J3. Similarly, r2 (resp. l2) and r3 (resp. l3) correspond to the

right (resp. left) generalized eigenvectors of J-1 and J1 respectively. Thus, the

generalized eigenvectors associated with the Jordan blocks can be defined as

r(J0)1 = r1, r

(J0)2 = r4, r

(J-1)1 = r2, r

(J1)1 = r3, l

(J0)1 = l1, l

(J0)2 = l4, l

(J-1)1 = l2, l

(J1)1 = l3.

(2.46)

Equivalently, we could also write the truncated Jordan normal form of F as

Junit =

1 0 0 1

0 -1 0 0

0 0 -1 0

0 0 0 1

. (2.47)

Since the columns of VR and VL are right and left generalized eigenvectors of F

77

corresponding to generalized eigenvalues of unit magnitude, they read [56]

r1 =

c1eR

0

0

0

r2 =

∗

c2eR

0

0

r3 =

∗

∗

c3eR

0

r4 =

∗

∗

∗

c4eR

l1 =

eLc1

∗

∗

∗

l2 =

0

eLc2

∗

∗

l3 =

0

0

eLc3

∗

l4 =

0

0

0

eLc4

(2.48)

where eR and eL are the χ2-dimensional left and right eigenvectors of the E cor-

responding to the eigenvalue 1 and the cj’s are some constants. The constant cj

can be set freely if rj and lj are eigenvectors (not generalized eigenvectors) of F .

However, in the calculation of WR and WL (defined in Eq. (2.35)), the generalized

eigenvectors ri and li of Eq. (2.48) are viewed as Υ×Υ matrices. They read

r1 =

c1eR 0

0 0

r2 =

∗ 0

c2eR 0

r3 =

∗ c3eR

∗ 0

r4 =

∗ ∗

∗ c4eR

l1 =

eL/c1 ∗∗ ∗

l2 =

0 ∗

eL/c2 ∗

l3 =

0 eL/c3

0 ∗

l4 =

0 0

0 eL/c4

(2.49)

where eR and eL are the right and left eigenvectors of the transfer matrix E, now

viewed as χ × χ matrices. Using Eqs. (2.46) and (2.39) (or directly Eqs. (2.47)

78

and (2.45)), WR and WL (whose components are defined in Eq. (2.38)) read

WR =

(r1 (-1)nr2 (-1)nr3 nr1 + r4

)WL =

(l1 + nl4 (-1)nl2 (-1)nl3 l4

). (2.50)

Using Eq. (2.37), we know that Runit and Lunit read

Runit = r1βrF 1 + (−1)nr2βrF 2 + (−1)nr3βrF 3 + (nr1 + r4)β

rF 4

Lunit = (l1 + nl4)βlF 1 + (−1)nl2βlF 2 + (−1)nl3βlF 3 + l4β

lF 4, (2.51)

where ri and li are Υ×Υ matrices defined in Eq. (2.49), and βrF i and βlF i are

the i-th components of the right and left modified boundary vectors.

Since we are mainly interested in the n → ∞ limit, we obtain ρred order by

order in n. Using Eq. (2.41), to order n2, the ρred which has the same spectrum

as the reduced density matrix (up to a global normalization factor), is given by

the product of O(n) terms from both Lunit and Runit in Eq. (2.51):

ρred = n2βlF 1βrF 4l4r

T1 +O(n). (2.52)

However, from Eq. (2.49), since l4rT1 = 0, ρred is a zero matrix at order n2.

Computing ρred to the next order n, ρred using Eq. (2.49), we obtain

ρred = nb14

eLeTR 0

∗ eLeTR

+O(1) (2.53)

Using Eq. (29), we know that eLeTR is nothing but the reduced density matrix

79

of the ground state. Since the ρred in Eq. (2.53) is block lower triangular, its

eigenvalues are those of its diagonal blocks. Thus, the entanglement spectrum,

given by the spectrum of ρred, of an MPO×MPS for a single-mode excitation is

two degenerate copies of the MPS entanglement spectrum, in the thermodynamic

limit (as n → ∞). We then immediately deduce that the entanglement entropy

is given by

S = SG + log 2 (2.54)

The extra log 2 entropy has an alternate interpretation as the Shannon entropy

due to the SMA quasiparticle being either in part A or part B of the system.

Thus, we have provided a proof that in the thermodynamic limit, single-mode

excitations have an entanglement spectrum that is two copies of the ground state

entanglement spectrum. Alternate derivations of the same result were obtained

in Refs. [139, 140].

In the AKLT models, the Arovas A and B states, and the spin-2 magnon of the

spin-1 AKLT model, the Arovas B states and the spin-2S magnon of the spin-

S AKLT model are all examples of single-mode excitations. While the Arovas

states are exact eigenstates only for periodic boundary conditions, it is reasonable

to believe that they are exact eigenstates for open boundary conditions too in

the thermodynamic limit. Thus, we expect their entanglement spectra to be two

degenerate copies of the ground state entanglement spectra in the thermodynamic

limit. While the entanglement spectra in the thermodynamic limit are the same

for all the single-mode excitations of the AKLT models, they differ in the nature

of their finite-size corrections.

80

2.5 Beyond Single-Mode Excitations

We now move on to the computation of the entanglement entropy of states that

are obtained by the application of multiple local operators on the ground state.

We focus on a concrete example in the 1D AKLT models, the tower of states of

Eq. (2.14). [55] We first focus on the state with two magnons (N = 2) and then

generalize the result to arbitrary N in the next section.

2.5.1 Jordan decomposition of the transfer matrix

For N = 2, the MPO MSS4 in Eq. (2.17) has a bond dimension χm = 3 and it

reads

MSS4 =

−1 −(S+)2S 0

0 1 (S+)2S

0 0 −1

. (2.55)

Consequently, using Eq. (2.19) and shorthand notations for the generalized trans-

fer matrices as

E+ ≡ E(S+)2S E− ≡ E(S−)2S E−+ ≡ E(S−)2S(S+)2S , (2.56)

81

the transfer matrix F can be written as a 9× 9 matrix:

F =

E E+ 0 E− E−+ 0 0 0 0

0 −E −E+ 0 −E− E−+ 0 0 0

0 0 E 0 0 E− 0 0 0

0 0 0 −E E+ 0 E− E−+ 0

0 0 0 0 E E+ 0 −E− E−+

0 0 0 0 0 −E 0 0 E−

0 0 0 0 0 0 E E+ 0

0 0 0 0 0 0 0 −E −E+

0 0 0 0 0 0 0 0 E

. (2.57)

The generalized eigenvalues of F that have magnitude 1 are due to the ±E blocks

on the diagonals of F . Thus, F has nine generalized eigenvalues of magnitude 1,

five (+1)’s and four (−1)’s. Using the property

E+eR = E−eR = 0 eTLE+ = eTLE− = 0 eTLE−+eR 6= 0 (2.58)

where eL and eR are the left and right eigenvectors of E corresponding to the

eigenvalue +1, it is possible to show that the largest generalized eigenvalues of

any two diagonal blocks in F belong to the same Jordan block if they are related

by an off-diagonal block E−+ in F [56]. Thus, for F , the truncated Jordan normal

82

form Junit of the generalized eigenvalues of largest magnitude reads

Junit =

1 0 0 0 1 0 0 0 0

0 −1 0 0 0 1 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 −1 0 0 0 1 0

0 0 0 0 1 0 0 0 1

0 0 0 0 0 −1 0 0 0

0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 −1 0

0 0 0 0 0 0 0 0 1

(2.59)

The forms of the right and left generalized eigenvectors corresponding to the

generalized eigenvalues in Junit can also be determined exactly. For example, the

left and right generalized eigenvectors corresponding to the fourth eigenvalue (−1)

on the diagonal of Junit in Eq. (2.59) read (when viewed as 3× 3 matrices)

r2,1 ≡ r4 =

∗ c1,2eR 0

∗ 0 0

∗ 0 0

l2,1 ≡ l4 =

0 eL

c1,2∗

0 ∗ ∗

0 ∗ ∗

(2.60)

where we have defined

rα,β ≡ r3(α−1)+β lα,β ≡ l3(α−1)+β (2.61)

to be the generalized eigenvectors of F corresponding to the generalized eigenvalue

of magnitude 1 and eR and eL are viewed as χ × χ matrices. Thus, in general,

83

the expression for the 3 × 3 rα,β (resp. lα,β) is obtained by filling in irrelevant

elements “∗”’s column-wise from top-to-bottom (resp. bottom-to-top) starting

from the top-left (resp. bottom-right) corner until the (α, β)-th element, which is

set to cα,βeR (resp. eL/cα,β). Using the structure of Junit in Eq. (2.59), we observe

that five Jordan blocks Jm, −2 ≤ m ≤ 2 are formed, that have generalized

eigenvalues (−1)m and consist of generalized eigenvectors rα,α+m and lα,α+m with

1−min(0,m) ≤ α ≤ 3−max(0,m).

2.5.2 General properties of Runit and Lunit

We now proceed to derive some general properties of Runit and Lunit that are

helpful in the calculation of ρred (see Eq. (2.41)). Since ρred is a sum products of

the form lα,βrTγ,δ (see Eq. (2.41)), using the forms of the generalized eigenvectors

lα,β and rα,β (for example Eq. (2.60)), we note the following properties:

lα,βrTγ,δ = 0 if β > δ, (2.62)

lα,βrTγ,β =

A if α > γ

A +∆(α, eLeTR) if α = γ

, (2.63)

where A represents a strictly lower-triangular matrix and ∆(α, x) is a diagonal

matrix with the α-th element on the diagonal equal to x. As we will see in the

next section, these properties are valid for any number of magnons N .

To compute ρred order by order in the length of the subsystem n, we need to

determine the factor of n that appears in front of the product lα,βrTγ,δ in ρred. We

first obtain the factors of n that accompany each of rα,β and lα,β in Runit and Lunit

respectively. Using Eqs. (2.39) and (2.37), when N = 2 the expression for Runit

84

reads

Runit =[((

n2

)r1,1 + nr2,2 + r3,3

)βrF 9 + (nr1,1 + r2,2)β

rF 5 + r1,1β

rF 1

]+(−1)n[(nr1,2 + r2,3)β

rF 8 + r1,2β

rF 4] + (−1)n[(nr2,1 + r3,2)β

rF 6 + r2,1β

rF 2]

+ [r1,3βrF 7] + [r3,1β

rF 3] , (2.64)

where terms in the same brackets come from the same Jordan block Jm. Similarly,

the expression for Lunit for N = 2 reads

Lunit =[((

n2

)l3,3 + nl2,2 + l1,1

)βlF 1 + (nl3,3 + l2,2)β

lF 5 + l3,3β

lF 9

]+(−1)n[(nl2,3 + l1,2)β

lF 4 + l2,3β

lF 8] + (−1)n[(nl3,2 + l2,1)β

lF 2 + l3,2β

lF 6]

+[l1,3β

lF 7

]+[l3,1β

lF 3

]. (2.65)

The structure of Eqs. (2.64) and (2.65) exemplify properties of R and L that are

valid for any value of N :

1. The largest combinatorial factors CRα,β and CL

α,β that multiply the right and

left generalized eigenvectors rα,β and lα,β in Runit and Lunit respectively read

(as a consequence of Eqs. (2.37) and (2.39))

CRα,β =

(n

N −max(α, β) + 1

), CL

α,β =

(n

min(α, β)− 1

). (2.66)

For example, the largest combinatorial factors to multiply r1,1 and l3,3 in

Eqs. (2.64) and (2.65) are CR1,1 =

(n2

)and CL

3,3 =(n2

)respectively.

2. The dominant term (with the largest factor of n) involving generalized eigen-

vectors of any given Jordan block are all multiplied by the same boundary

85

vector component in the expression for Lunit and Runit. This is derived using

Eqs. (2.37) and (2.39). For example, r1,1, r2,2 and r3,3 (resp. l1,1, l2,2 and l3,3)

are all associated with the same Jordan block (J0), and the largest factors

of n that multiply them are(n2

)βrF 9, nβrF 9 and βrF 9 (resp. βlF 1, nβlF 1, and(

n2

)βlF 1). That is, the dominant terms involving these generalized eigenvec-

tors are all multiplied by the same boundary vector component βrF 9 (resp.

βlF 1) in Runit (resp. Lunit).

3. All the terms in Eq. (2.63) associated with a given Jordan block are mul-

tiplied by λn, where λ is the eigenvalue associated with the Jordan block

involved (here either (+1) or (−1)). This is seen in Eq. (2.39).

Using CLα,β and CR

α,β of Eq. (2.66), we can directly compute ρred (defined in

Eq. (28)) order by order in n. Note that

CLα,βC

Rγ,δ ∼ O

(nN+min(α,β)−max(γ,δ)) . (2.67)

Using Eq. (2.67), we note that any term of order strictly greater than nN requires

min(α, β) > max(γ, δ), which necessarily implies β > δ. Since all products lα,βrTγ,δvanish (using Eq. (2.62)), the dominant non-vanishing terms appear at order nN

or smaller. Directly from Eq. (2.67), if β < δ, β < γ, α < γ or α < δ, the product

CLα,βC

Rγ,δ necessarily has a smaller order than nN . Thus, at order nN , we obtain

products that satisfy α ≥ γ, α ≥ δ, β ≥ δ and β ≥ γ. The products with β > δ

vanish (using Eq. (2.62)) and products with α > γ give rise to lower triangular

terms (using Eq. (2.63)); and they do not contribute to the eigenvalues of ρred

when no upper triangular terms are present. We thus deduce that the products

86

that determine the spectrum of ρred (and hence the entanglement spectrum) at

leading order in n satisfy β = δ, α = γ, α ≥ δ and β ≥ γ; and consequently,

α = β = γ = δ. Furthermore, since all the rα,α’s and lα,α’s belong to the largest

Jordan block with eigenvalue +1, all the products lα,αrTα,α are multiplied with the

same modified boundary vector components.

Indeed, these arguments can be verified using the exact form of ρred at order n2

using Lunit and Runit in Eqs. (2.64) and (2.65). Thus, at order n2, ρred reads

ρred = b1,9

(n2

)eLe

TR 0 0

∗ n2eLeTR 0

∗ ∗(n2

)eLe

TR

+O(n)

≈ n2b1,9

12eLe

TR 0 0

∗ eLeTR 0

∗ ∗ 12eLe

TR

+O(n), (2.68)

where we have used(n2

)≈ n2

2, an approximation that is exact as n → ∞. The

entanglement spectrum of two magnons on the ground state is thus three copies of

the ground state entanglement spectrum. The three copies are however, separated

into one non-degenerate and two degenerate copies.

2.6 Tower of States

We now move on to the calculation of the entanglement spectra for the AKLT

tower of states with N > 2 magnons on the ground state. The expression for the

MPO MSS2Nfor the tower of states operator has a bond dimension χm = N + 1

and is shown in Eq. (2.17). Several results in this section are a straightforward

87

generalization of results in the previous section.

2.6.1 Jordan decomposition of the transfer matrix

Analogous to Eq. (2.57), the transfer matrix F for arbitrary N can be written as

a (N + 1)× (N + 1) block upper triangular matrix, with χ× χ blocks. Thus, the

generalized eigenvectors of F for a general N have inherited a structure as those

in Eq. (2.60). The right and left generalized eigenvectors rα,β ≡ r(N+1)(α−1)+β and

lα,β ≡ l(N+1)(α−1)+β have the forms (when viewed as (N + 1)× (N + 1) matrices),

rα,β =

∗ · · · · · · ∗ 0 · · · 0

... . . . . . . ... ... . . . ...

∗ · · · · · · ∗ ... . . . ...

∗ · · · ∗ cα,βeR 0 · · · 0

... . . . ... 0 · · · · · · 0

... . . . ... ... . . . . . . ...

∗ · · · ∗ 0 · · · · · · 0

, lα,β =

0 · · · · · · 0 ∗ · · · ∗... . . . . . . ... ... . . . ...

0 · · · · · · 0... . . . ...

0 · · · 0 eLcα,β

∗ · · · ∗... . . . ... ∗ · · · · · · ∗... . . . ... ... . . . . . . ...

0 · · · 0 ∗ · · · · · · ∗

,(2.69)

where the (α, β)-th element in rα,β and lα,β are proportional to eR and eL re-

spectively. Since the off-diagonal blocks of F have the same structure as those

in Eq. (2.57) (because the structures of the MPOs MSS2 and MSS2Nare the

same), the Jordan normal form is similar to the N = 2 case. That is, we

obtain (2N + 1) Jordan blocks Jm, −N ≤ m ≤ N , that correspond to an

eigenvalue (−1)m and consist of generalized eigenvectors rα,α+m and lα,α+m with

1−min(0,m) ≤ α ≤ N + 1−max(0,m).

As pointed out in Sec. 2.5.2, the properties observed there are valid for all N .

88

Thus, using CRα,β and CL

α,β, ρred can be constructed order by order in n. However,

for arbitrary N , we can study two types of limits (i) n → ∞, N finite, and (ii)

n→∞, N →∞, N/n→ const. > 0. Since n = L/2, N is the number of magnons

in the state |SS2N〉, and the state has an energy E = 2N , the energy density of

the state we are studying is E/L = N/n. The limits (i) and (ii) thus correspond

to zero and finite energy density excitations respectively.

2.6.2 Zero density excitations

In the limit where N is finite as n→∞, we can use the approximation

(n

N

)≈ nN

N !, (2.70)

which is asymptotically exact. Thus, the product of combinatorial factors can be

classified by order in n. Since the structure of the generalized eigenvectors lα,β

and rα,β in Eq. (2.69) are the same as the N = 2 case in the previous section,

properties Eqs. (2.62) and (2.63) are valid here. Using the arguments following

Eq. (2.67) in Sec. 2.5.2, the first non-vanishing term appears at order nN , and the

expression for ρred reads

ρred = b1,(N+1)2

N∑α=0

(n

α

)(n

N − α

)lα,αr

Tα,α + A +O(nN−1)

≈ nNb1,(N+1)2

N∑α=0

1

α!(N − α)!lα,αr

Tα,α + A +O(nN−1)

(2.71)

89

where A represents strictly lower triangular matrices. Using Eq. (2.63), to leading

order in n, we obtain the unnormalized density matrix:

ρred = nNb1,(N+1)2

eLeTR

N !0!0 . . . . . . 0

∗ eLeTR

(N−1)!1!

. . . . . . ...... . . . . . . . . . ...... . . . . . . eLe

TR

1!(N−1)!0

∗ . . . . . . ∗ eLeTR

0!N !

(2.72)

where eLeTR is the ground state reduced density matrix. Since eLeTR for the spin-S

AKLT model has (S + 1) degenerate levels (see Eq. (2.6)), after normalizing ρred,

the entanglement spectrum has (N + 1) copies of (S + 1) degenerate levels, and

each (S + 1)-multiplet reads

λα =1

2N(S + 1)

(N

α

)0 ≤ α ≤ N. (2.73)

The trace of ρred can indeed be verified to be 1. The entanglement entropy is thus

S = −Tr [ρred log ρred] = −(S + 1)N∑α=0

λα log λα = SG +N log 2− 1

2N

N∑α=0

(N

α

)log

(N

α

)∼ SG +

1

2log

(πN

2

)for large N (2.74)

where SG = log(S + 1), the entanglement entropy of the spin-S AKLT ground

state. The large N behavior in Eq. (2.74) is derived from using Stirling’s approx-

imation followed by a saddle point approximation [56]. For N = 1, we recover the

Single-Mode Approximation result of Eq. (2.54). Furthermore, note that O(nN−1)

and lower order corrections to ρred in Eq. (2.72) are typically not lower triangu-

90

lar matrices. Thus, the replica structure of ρred breaks at any finite n, giving a

particular structure to the finite-size corrections.

2.6.3 Finite density excitations

We now proceed to the case where the excited state has a finite energy den-

sity, corresponding to a finite density of magnons on the ground state. That

is, E/L = N/n > 0. For a large enough N , approximation Eq. (2.70) breaks

down. Nevertheless, since the MPO for |SS2N〉 and the MPS for the ground

state of the spin-S AKLT model have bond dimensions of χm = (N + 1) and

χ = (S + 1) respectively, the MPO × MPS for |SS2N〉 has a bond dimension

χχm = (S + 1)(N + 1), i.e. it grows linearly in N . Consequently, using Eq. (32),

the entanglement entropy of |SS2N〉 is bounded by

S ≤ log(χχm) = log[(S + 1)(N + 1)]. (2.75)

Using Eqs. (2.74) and (2.75), we would be tempted to find a stronger bound or an

asymptotic expression for the entanglement entropy in the finite density limit. In-

deed, we expect this entanglement entropy to have the form S ∼ P logN , where

P is some constant. Without the approximation of Eq. (2.70), terms that are

weighted by the combinatorial factor(na

)(nk−a

)do not necessarily suppress the

terms that appear with a factor(na

)(n

k−a−b

), where k, a and b are some positive

integers. This invalidates an expansion in orders of n such as Eq. (2.71). Conse-

quently the lower triangular structure of ρred (see Eq. (2.72)) breaks down. Hence,

it is not clear if the expression for the entanglement entropy of Eq. (2.74) survives

in the finite density regime.

91

2 4 6 8 10 12 14

E

0.5

0.6

0.7

0.8

S/(L/2

log

3)

(a) L = 14

α = 0.0

2 4 6 8 10 12 14 16

E

0.5

0.6

0.7

0.8

S/(L/2

log

3)

(b) L = 16

α = 0.0

2 4 6 8 10 12 14

E

0.5

0.6

0.7

0.8

S/(L/2

log

3)

(c) L = 14

α = -0.025

2 4 6 8 10 12 14 16

E

0.5

0.6

0.7

0.8

S/(L/2

log

3)

(d) L = 16

α = -0.025

2 4 6 8 10 12 14

E

0.5

0.6

0.7

0.8

S/(L/2

log

3)

(e) L = 14

α = -0.05

2 4 6 8 10 12 14 16

E

0.5

0.6

0.7

0.8

S/(L/2

log

3)

(f) L = 16

α = -0.05

Figure 2.3: The normalized entanglement entropy S/((L/2) log 3) for the eigen-states of the Hamiltonian with energy E Eq. (2.76) in the quantum number sector(s, Sz, k, I, Pz) = (6, 0, π,−1,+1), where the quantum numbers respectively cor-respond to the total spin, the projection of the total spin along the z direction,momentum, inversion and spin-flip symmetries. [55] Panels (a) and (b) show theentropy at the AKLT point. This sector has a single exact state |S6〉 that belongsto the tower of states, which exhibits a sharp dip at E = 6. Panels (c) and (d)show the entropy for α = −0.025, where remnants of the low entropy states areseen. Panels (e) and (f) show the entropy in the same sector for α = −0.05.

2.7 Implications for ETH

In the previous chapter, we conjectured and provided numerical evidence that in

the thermodynamic limit some states of the tower of states are in the bulk of the

spectrum, i.e. in a region of finite density of states of their own quantum number

92

sector. Furthermore, we showed that the AKLT model is non-integrable, i.e. it

exhibits Gaussian Orthogonal Ensemble (GOE) level statistics. As discussed in

Sec. 0.1 in Chapter 0, according to the Eigenstate Thermalization Hypothesis

(ETH), typical states in the bulk of the spectrum look thermal [15, 16, 18]. That

is, the entanglement entropy of any such states exhibits a volume law scaling,

S ∝ L. A strong form of the ETH conjuctures that all states in a region of finite

density of states of the same quantum number sector look thermal [96, 97].

In the spin-S AKLT tower of states, for a state with a finite density of magnons,

using Eq. (2.75), S ∝ logL. The logL scaling of the entanglement entropy in

Eq. (2.75) is thus a clear violation of the strong ETH. The atypical behavior of

the tower of states is illustrated in Fig. 2.3. In Figs. 2.3a and 2.3b, we plot the

entanglement entropy of all the states in a given quantum number sector for two

system sizes L = 14 and L = 16 at the AKLT point. The dip of the entanglement

entropy at energy E = 6 corresponds to the state |S6〉, which clearly violates the

trends of entanglement entropy within its own quantum number sector. The dip

persists for L = 16, the largest system size accessible to exact diagonalization.

These states are thus the first examples of what are now known as “quantum

many-body scars” [6, 67, 68, 138]. One might wonder if such a violation of ETH

is generic in nature, i.e., if these states have a sub-thermal entanglement entropy

even when the Hamiltonian is perturbed away from the AKLT point. We explore

this using the Hamiltonian

Hα =L∑i=1

(1

3+

1

2Si · Si+1 +

(1

6+α

2

)(Si · Si+1)

2

)(2.76)

where α = 0 corresponds to the Hamiltonian of the AKLT model. We find that

93

the dip in the entanglement entropy is stable up to a value of α = −0.025, as

shown in Figs. 2.3c and 2.3d. However, we cannot exclude that the range of α

where we observe this low entanglement in the bulk of spectrum, will go to zero

in the thermodynamic limit (as observed for α = −0.05 in Figs. 2.3e and 2.3f).

Since the number of states that belong to the tower of states grows only poly-

nomially in L, the set of ETH violating states has a measure zero. Thus, the

existence of these sub-thermal states do not preclude the weak ETH, which states

almost all eigenstates in a region of finite density of states look thermal.

2.8 Entanglement Spectra Degenarcies and Finite-Size

Effects

We now move on to describe the constraints that the AKLT Hamiltonian symme-

tries on the entanglement spectra of the exact excited states.

2.8.1 Symmetries of MPS and symmetry protected topological phases

We first briefly review the action of symmetries on an MPS, the concept of Symme-

try Protected Topological (SPT) phases in 1D, and their connections to degenera-

cies in the entanglement spectrum [120,121,135,164]. A state |ψ〉 that is invariant

under any symmetry (that has a local action on an MPS) admits an MPS rep-

resentation that transforms under the particular symmetry as [113, 120, 164, 165]

u(A[m]) = eiθUA[m]U †, (2.77)

where u is the symmetry operator that transforms the MPS, U is a unitary matrix

that acts on the ancilla, and eiθ is an arbitrary phase. We now discuss various

94

useful symmetries that are relevant to the AKLT models.

Since the inversion symmetry flips the chain of length L (and hence the MPS

representation of the state) by interchanging sites i and L−i, the site-independent

MPS of the transformed state is the same as the MPS of the original state read

from right to left in Eq. (18). Consequently, the site-independent MPS transforms

under inversion as [120]

uI(A[m]) = (A[m])T = eiθIU †

IA[m]UI (2.78)

In Ref. [120], it was shown that for an MPS A in the canonical form, the UI

matrices should satisfy UIU∗I = ±1. This leads to a double degeneracy of each

level of the entanglement spectrum [56,120]. The origin of the degeneracy can be

traced back to the existence of symmetry protected edge modes at the ends of the

chain and the SPT phase.

Time-reversal, by virtue of being an anti-unitary operation, acts on the MPS

as

uT (A[m]) =

∑n

TmnA[n] = eiθTU †TA

[m]UT (2.79)

where Tmn =(eiπS

yp)mnK, where K is the complex conjugation operator and Syp

acts on the physical index. The two classes of UT matrices are again UTU∗T = ±1,

with UTU∗T = −1 indicating an SPT phase [120].

In the case of Z2 × Z2 spin-rotation symmetries (π-rotations about x and z

axes), the MPS transforms under the symmetries as

uσ(A[m]) =

∑n

RσmnA[n] = eiθσU †

σA[m]Uσ (2.80)

95

where Rσmn =(eiπS

σp)mn

, σ = x, z and Sσp acts on the physical index. The two

classes of Uσ are the ones that satisfy UxUzU †xU

†z = ±1, where UxU∗

x = UzU∗z = 1.

Thus the classes of matrices can be written as (UxUz)(UxUz)∗ = ±1.

In each of the cases above, we refer to the transformations with positive and

negative signs as linear and projective transformations respectively. Since the

conditions of SPT order for the symmetry groups are of the form UU∗ = −1,

where U is unitary, U should be χ × χ anti-symmetric matrix. If χ is odd, 0

is an eigenvalue of U , contradicting the fact that U is unitary. Thus, protected

degeneracies cannot exist due to the symmetries we have discussed if the bond

dimension of the MPS representation in the canonical form is odd.

The spin-1 AKLT ground state MPS Eq. (2.1) satisfies Eqs. (2.78), (2.79) and

(2.80) with UI = UT = iσy, Ux = σx and Uz = σz. Thus the entanglement

spectrum of the spin-1 AKLT ground state is degenerate. This analysis can be

extended straight forwardly to a spin-S AKLT model groundstate. Since even

S AKLT ground states have an odd bond dimension, they do not have SPT

order nor a doubly degenerate entanglement spectrum. For odd S, the operators

UI = UT = eiπSya , Ux = eiπS

xa and Uz = eiπS

za , where Sσa (σ = x, y, z) are spin-S/2

operators that act on the ancilla, satisfy Eqs. (2.78), (2.79) and (2.80) respectively.

Since these matrices satisfy

UIU∗I = UTU

∗T = (UxUz)(UxUz)

∗ = (−1)S1, (2.81)

all odd-S AKLT chains have SPT order and a doubly degenerate entanglement

spectrum whereas even-S chains do not.

96

2.8.2 Symmetries of MPO

For any Hamiltonian that is invariant under certain symmetries, each of eigen-

states are labelled by quantum numbers corresponding to a maximal set of com-

muting symmetries. As shown in the previous section, the AKLT ground states

are invariant under inversion, time-reversal, and Z2 × Z2 rotation symmetries.

However, some of the excited states we consider are not invariant under the said

symmetries. For example, the tower of states we have consider have Sz 6= 0, and

are not invariant under time-reversal or Z2×Z2 symmetries but they are invariant

under inversion symmetry.

When an excited state is invariant under a certain symmetry, it can trivially be

expressed in terms of an operator invariant under the same symmetry acting on

the ground state. Thus, analogous to Eq. (2.77), under a symmetry u, the MPO

of such an operator should transform as

u(M [mn]) = eiθΣ†M [mn]Σ. (2.82)

where u acts on the physical indices of the MPO and Σ on the ancilla.

We first discuss the symmetries that we discussed with regard to MPS in

Sec. 2.8.1, i.e., inversion, time-reversal and Z2×Z2 rotation. The actions of these

symmetries on an MPO are similar to the actions on the MPS. Corresponding to

97

inversion, time-reversal, and Z2 × Z2 rotation symmetries, we obtain

uI(M[mn]) = (M [mn])T = eiθIΣ†

IM[mn]ΣI ,

uT (M[mn]) =

∑l,k

TmlM [lk]T †kn = eiθTΣ†

TM[mn]ΣT ,

uσ(M[mn]) =

∑l,k

RσmlM[lk]R†

σkn = eiθσΣ†σM

[mn]Σσ, (2.83)

where Tmn =(eiπS

yp)m,nK, Rσmn =

(eiπS

σp)mn

, σ = x, z act on the physical index

of the MPO. In each of these cases, the auxiliary indices of the MPO transform

under the ΣI , ΣT , Σx, Σz matrices under the various symmetries respectively.

Similar to the case of an MPS, we could have MPOs that transform in two dis-

tinct ways ΣIΣ∗I = ±1, ΣTΣ

∗T = ±1 and (ΣxΣz)(ΣxΣz)

∗ = ±1. We refer to the

transformation with the positive and negative signs as linear and projective MPO

transformations respectively. Thus, under physical symmetries, if an MPS trans-

forms on the ancilla under U , and an MPO transforms under Σ, the MPO × MPS

transforms on the ancilla under U ⊗Σ. As a consequence, if an MPO transforms

projectively (resp. linearly), the MPO × MPS transforms in a different (resp. the

same) way as the MPS.

It is straightforward to compute the transformation matrices for various MPOs

corresponding to exact excited states in the AKLT model. For example, the MPO

corresponding to the Arovas A operator (see Eq. (2.7)) transforms linearly under

inversion, time-reversal and Z2×Z2 rotation symmetries. The Arovas B operator

transforms projectively under inversion, linearly under time-reversal and rotation

symmetries. The tower of states operator transforms projectively and linearly

under inversion symmetry for odd and even N respectively. Note that we do not

98

claim any topological protection of these states. Indeed, they have a degenerate

largest eigenvalue of the transfer matrix, leading to long-range correlations that

do not decay exponentially.

2.8.3 Examples from the AKLT Models

We now proceed to describe the finite-size effects and symmetry-protected de-

generacies in the entanglement spectra of the exact excited states of the AKLT

models. Since the exact entanglement spectra depend on the configuration of the

free boundary spins, we freeze them to their highest weight states. Such a bound-

ary configuration is inversion symmetric, although it violates time-reversal and

Z2 × Z2 rotation symmetries (on the edges only).

Spin-S AKLT ground states

As described in Sec. 2.8.1, the entanglement spectrum of the spin-S AKLT ground

state consists of (S+1) degenerate levels in the thermodynamic limit. Generically,

such a degeneracy between (S+1) levels is broken for a finite system. However, as

shown in the thermodynamic limit in Ref. [120] and for a finite system in Ref. [56],

the entanglement spectrum is always doubly degenerate when symmetries act

projectively. Thus, for odd S, since inversion, time-reversal and Z2 × Z2 act

projectively (see Eq. (2.81)), the entanglement spectrum consists of (S + 1)/2

exactly degenerate doublets. For even-S, the entanglement spectrum need not

consist of degenerate levels for generic configurations of boundary spins, though

some levels can be degenerate for particular choices of the boundary spins. While

the exact form of the splitting between the entanglement spectrum levels depends

on the configuration of the boundary spins, we find that it is exponentially small

99

in the system size.

Spin-1 AKLT tower of states

We first describe the entanglement spectrum of the spin-2 magnon state of spin-1

AKLT model, |S2〉. In Sec. 2.4, we have seen that the entanglement spectrum of

such a state consists of two copies of the ground state entanglement spectrum.

For a finite n, using an explicit computation of ρred using the methods described

in Sec. 2.3.2, with MPS boundary vectors of Eq. (2.3), the four normalized eigen-

values of ρred read

2×(

n

4n− 3,3− 2n

6− 8n

), (2.84)

where 2× indicates two copies. In Eq. (2.84), we have ignored exponentially small

finite-size splitting to obtain a closed form expression. The two degenerate copies

of the entanglement spectrum thus split into two doublets that have an O(1/n)

(power-law) splitting. Similarly, the six eigenvalues of ρred for |S4〉 read

2×(

4n2 − 22n+ 27

32n2 − 88n+ 54,

2n2 − 5n

16n2 − 44n+ 27,

4n2 − 6n

16n2 − 44n+ 27

)(2.85)

This is consistent with the n → ∞ behavior calculated in Sec. 2.6.2, i.e. the

entanglement spectrum is composed of three copies of the ground state split into

three doublets, two of which are degenerate in the thermodynamic limit at half

the entanglement energy of the other. The doublets that are degenerate in the

thermodynamic limit have an O(1/n) finite-size splitting between them.

More generically, we observe the following pattern in the entanglement spectrum

of |S2N〉. The (N +1) copies of the ground state split into (N +1) doublets, some

of which are separated by O(1/N) in the thermodynamic limit. The pairs of

100

doublets that are degenerate in the thermodynamic limit have a power-law finite

size splitting of O(1/n). A schematic plot of the entanglement spectra of the

tower of states is shown in Fig. 2.4.

We now distinguish between exact degeneracies and exponential finite-size split-

tings. As shown in Sec. 2.8.2, the MPO for the tower of states transforms projec-

tively (resp. linearly) under inversion symmetry if N is odd (resp. even). Since

the spin-1 AKLT ground state transforms projectively under inversion, the MPO

× MPS transforms projectively (resp. linearly) under inversion symmetry if N is

even (resp. odd). While the proof for double degeneracy due to projective repre-

sentations in Ref. [120] relied on the uniqueness of the largest eigenvalue of the

transfer matrix of the MPS, in Ref. [56] we show the existence of the degeneracy

in the mid-cut entanglement spectrum for a finite system irrespective of the struc-

ture of the transfer matrix. Consequently, we observe exact degeneracies of the

doublets for even N and exponential finite-size splittings within the doublets for

odd N . This effect is schematically shown in Fig. 2.4. The exponential splitting

happens for generic symmetry-preserving configurations of the boundary spins,

though certain configurations of boundary spins lead to “accidental” degeneracies

in the entanglement spectrum.

Spin-S AKLT tower of states

Similar to the spin-1 AKLT tower of states, we compute the exact entanglement

spectra for the spin-S tower of states of Ref. [ [55]]. We start with S = 2. The

spectrum of ρred for the state |2S2〉 (obtained via a direct computation) has six

101

Figure 2.4: Schematic depiction of the entanglement spectra of spin-1 AKLTtower of states |S2N〉 (left) and spin-2 AKLT tower of states |2S2N〉 (right).The almost-degenerate levels shown in red have an exponential finite-size splittingwhereas the black doublets are exactly degenerate. Power-law finite-size splittingsare depicted by black two-headed arrows and constants by blue two-headed arrows.

eigenvalues that read

2×(

9n+ 28

84 + 54n,

9n+ 4

84 + 54n,9n+ 10

84 + 54n

). (2.86)

Similar to the spin-1 case, we note that the two copies of the ground state en-

tanglement spectrum split into three doublets that are separated by an O(1/n)

finite-size splitting. For the state |2S4〉, the eigenvalues of ρred read (ignoring

exponentially small splitting)

2×(

27n2+117n−806(54n2+117n−40)

, 27n2−27n−1046(54n2+117n−40)

), 2×

(27n2+9n−128

6(54n2+117n−40), (9n+28)(9n+10)9(54n2+117n−40)

),

1×(

(9n+4)2

9(54n2+117n−40)

)(2.87)

102

Thus, we find that the nine levels due to the three copies of the ground state

entanglement spectrum split into four doublets and one singlet. Two of the copies

of the ground state entanglement spectrum are degenerate in the thermodynamic

limit, and at a finite size, these six entanglement levels split into three doublets

that have an O(1/n) splitting.

We numerically observe that a similar pattern holds true for arbitrary S. For

the state |SS2N〉, the (N + 1) copies of the ground state entanglement spectrum

(that consists of (S + 1) levels) splits into doublets and singlets. If S is odd, we

obtain (S+1)/2 doublets and if S is even, we obtain S/2 doublets and one singlet.

The doublets and singlets that are degenerate in the thermodynamic limit have

an O(1/n) finite-size splitting. Furthermore, as shown in Sec. 2.8.2, the MPO

for the tower of states transforms projectively (resp. linearly) under inversion

symmetry if N is odd (resp. even). Consequently, the MPO × MPS transforms

projectively (resp. linearly) under inversion symmetry if (N + S) is odd (resp.

even). Indeed, similar to the spin-1 AKLT tower of states, we find exactly degen-

erate doublets in the entanglement spectrum for arbitrary symmetry-preserving

boundary conditions when the MPO × MPS transforms projectively (i.e. when

(N +S) is odd). If (N +S) is even, we find that for generic symmetry-preserving

boundary conditions, we obtain an exponential finite-size splitting between the

doublets that are degenerate in the thermodynamic limit.

103

Spin-1 Arovas states

For the spin-1 Arovas A state, via a direct computation, we find that the eight

eigenvalues of ρred read

2×

(n+ 3 + 2

√2(1 + n)

4n+ 14,n+ 3− 2

√2(1 + n)

4n+ 14

), 4×

(1

8n+ 28

). (2.88)

Thus, similar to the spin-2 magnon, we obtain two copies of the ground state en-

tanglement spectrum that splits into two doublets that have an O(1/√n) splitting

between them. In addition, we obtain 4 entanglement levels that are of O(1/n).

As mentioned in Sec. 2.8.2, the Arovas A MPO transforms linearly under inversion,

time-reversal and Z2 × Z2 symmetries. Consequently, the MPO × MPS trans-

forms projectively and all the doublets are exactly degenerate for a finite system.

While we were not able to obtain a closed-form expression for the entanglement

spectra of the spin-1 and spin-2 Arovas B states, we numerically observe similar

phenomenology as the Arovas A and the spin-2S magnon entanglement spectra,

although the magnitude of the finite-size splittings (O(1/√n) versus O(1/n)) are

not clear.

2.9 Conclusion

We have computed the entanglement spectra of the exact excited states of the

AKLT models that were derived in Chapter 1. To achieve this, we expressed

the states as MPO × MPS’ and developed a general formalism to compute the

entanglement spectra of states using the Jordan normal form of the MPO ×

MPS transfer matrix. We first exemplified our method by reproducing existing

104

results on single-mode excitations: we show that their entanglement spectra in

the thermodynamic limit consist of two copies of the ground state entanglement

spectrum. The low-lying exact excited states of the AKLT model such as the

Arovas states and the spin-2S magnon states for the spin-S AKLT chain fall into

this category. For single-mode excitations, our method is exactly equivalent to the

tangent-space and related methods developed to numerically as well as analytically

probe low-energy excited states in the MPS formalism [139,142, 159, 166].

We then generalized our method to states with multiple magnons, that are

beyond single-mode excitations (“double tangent space” [167] and beyond). This

allowed us to obtain the exact expression for the entanglement spectra for the spin-

S AKLT tower of states for a zero density of magnons in the thermodynamic limit.

We showed that the entanglement spectrum of the N -th state of the tower consists

of (N + 1) copies of the ground state entanglement spectrum, not all degenerate.

Apart from the specific Jordan block structure derived for the special AKLT

tower of states, our method to obtain the entanglement spectrum was completely

general. In particular, it applies to states of the form ON |ψ〉, where O is any

translation invariant operator and |ψ〉 is a state that admits a site-independent

MPS representation. For the AKLT tower of states, we also showed that the

replica structure of the entanglement spectra of the tower of states persists in

the thermodynamic limit only for states at a zero energy density, conforming with

folklore that only low energy excitations resemble the ground state. An interesting

problem is to prove this on general grounds for excited states in integrable and/or

non-integrable models.

We also studied finite-size effects in the entanglement spectra of these states

and showed a universal power-law splitting between the different copies of the

105

ground state. We identified exact degeneracies and exponential splittings based

on projective versus linear transformations of the MPO × MPS at a finite size.

While protected exact degeneracies in the entanglement spectrum of excited states

are reminiscent of SPT phases for the ground states, it is unclear if these have a

topological origin in the excited states, given that excited states do not have a

protecting gap.

We emphasized that the states of the tower have an entanglement entropy that

scales as S ∝ logL, which is incompatible with strong ETH, if these states indeed

exist in the bulk of the energy spectrum as suggested in Chapter 1. Further, we

showed that the violation of ETH seems to persist for SU(2) symmetric spin-1

Hamiltonians slightly away from the AKLT point. However, a systematic numer-

ical study of these low-entropy states away from the AKLT point is necessitated,

with and without breaking the SU(2) symmetry. These special states thus pro-

vide analytically tractable examples of “quantum many-body scars”, described in

Refs. [68, 138]. While such anomalous eigenstates are known to exist in single-

particle chaotic systems, very few examples are known in many-body quantum

systems [168, 169].

106

3Quantum Scars from Matrix Product

States

3.1 Introduction

Subsequently to the discovery of the AKLT model, similar examples of exact ETH-

violating eigenstates were discovered in a variety of non-integrable models. Some

non-integrable systems exhibit a few solvable eigenstates [75, 170], whereas exact

107

towers of states embedded in a thermal spectrum were discovered in a variety

of models, for example in the spin-1 XY models [57, 59] and a spin-1/2 domain-

wall conserving model [58, 61]. In some of these models, the quantum scars can

be understood using a formalism developed by Ref. [53], where ETH-violating

eigenstates can be embedded systematically in the middle of an ETH-satisfying

spectrum. However, it has not been clear if some other models - for example the

AKLT chain - are isolated scarred points in the space of Hamiltonians or if they

are part of a much larger family of quantum scarred Hamiltonians, although work

in Ref. [61] has shed light on this question for the AKLT chain.

Given that the ground state of the AKLT chain [54] is also a paradigmatic ex-

ample of a Matrix Product State (MPS), it is natural to wonder if the powerful

tools developed in the context of MPS [108, 110, 111] can be used to understand

the exact excited states in the AKLT chain. In this chapter, we show that such

a connection indeed exists, and we provide a general formalism for construct-

ing quantum scarred Hamiltonians starting from an MPS wavefunction. Given an

MPS wavefunction, the so-called parent Hamiltonian construction provides a fam-

ily of Hamiltonians for which the said MPS is an eigenstate. In the large family

of such parent Hamiltonians, we illustrate a method to look systematically for the

subfamilies of Hamiltonians with quantum scars. Using this approach, we recover

the analytical examples of quantum scars of the AKLT chain and generalize them

in three directions. First, we obtain a 6-parameter family of nearest-neighbor

Hamiltonians that all have the AKLT tower of states as exact eigenstates. Sec-

ond, we start with a generalization of the AKLT MPS and obtain a class of

Hamiltonians with new towers of exact eigenstates. Using this generalization, we

also show that the AKLT chain can be continuously deformed to the (integrable)

108

spin-1 biquadratic model, while preserving the quantum scars. Finally, we use

our formalism to show examples of new types of quantum scars in a Potts model

perturbed to have a U(1) symmetry and exact ground states [171], and we discuss

generalizations therein.

This chapter is organized as follows. In Sec. 3.2, we review the basic concepts

of MPS and quasiparticle excitations in the MPS language used in the rest of

the paper. In Sec. 3.3, we review the construction of parent Hamiltonian of an

MPS ground state using the AKLT chain as an example. In Sec. 3.4, we give the

main result of the paper, the extension of the parent Hamiltonian construction to

include a tower of states composed of single-site quasiparticles. We illustrate this

method by obtaining a family of Hamiltonians for which the AKLT tower of states

remain eigenstates. In Sec. 3.5, we use our formalism to obtain a new family of

quantum scarred models starting from a generalized AKLT MPS. We construct

a continuous scar-preserving path from the AKLT chain to the integrable spin-1

biquadratic model. Further, in Sec. 3.6, we discuss the extension of our formalism

to a tower of two-site quasiparticles, and we show that the U(1)-invariant per-

turbed Potts model of Ref. [171] exhibits such a tower of states. We present our

conclusions in Sec. 3.7.

3.2 MPS description of Quasiparticles

In this section, we briefly review MPS description of the ground state and quasi-

particle excited states and introduce notations that we use in this chapter.

109

3.2.1 Ground State

Consider a one-dimensional quantum chain with a d-dimensional Hilbert space on

each of the L sites. The many-body basis of the system is labelled by |m1,m2, · · · ,mL〉,

where mj runs over a basis of the single-site Hilbert space. A wavefunction |ψ〉

on such a system is an Matrix Product State (MPS) if its decomposition in this

basis reads [111]

|ψ〉 =∑mj

Tr[A

[m1]1 A

[m2]2 · · ·A[mL]

L

]|m1,m2, · · ·mL〉. (3.1)

Here A[mj ]j is a χ×χ matrix, χ being the bond dimension of the MPS and thus the

Aj’s are d×χ×χ tensors. The trace arises from the periodic boundary conditions

we impose. In this chapter, we use the following graphical and shorthand notations

to represent such a wavefunction

|ψ〉 = = |[A1A2 · · ·AL−1AL]〉 . (3.2)

In Eq. (3.2), we use the brackets [ ] to indicate that the auxiliary indices at the

ends have been contracted. It is also sometimes useful to address segments of the

wavefunction |ψ〉 of Eq. (3.2), for which we use the shorthand notation without

the brackets [ ], for example

|A1A2〉 = ≡∑m1,m2

A[m1]1 A

[m2]2 |m1,m2〉. (3.3)

In Eq. (3.3), A[m1]1 A

[m2]2 is a χ× χ matrix for given values of m1 and m2.

110

Although the tensors Aj in Eq. (3.1) can be site-dependent, a translation-

invariant wavefunction can always be represented by an MPS with a site-independent

tensor A [111]. That is, any translation invariant MPS wavefunction |ψA〉 can be

represented as

|ψA〉 = |[AA · · ·A]〉 , (3.4)

where we have used the shorthand notation of Eq. (3.2).

3.2.2 Quasiparticle Excitations

As we discussed in Chapter 2, MPSs can also be used to efficiently describe quasi-

particle excitations above the ground state. Here, we directly work with the MPS

instead of employing MPOs. These techniques were pioneered by works on so-

called tangent space methods [142,159,172]. A single-site quasiparticle excitation

with momentum k on top of the MPS state |ψA〉 is given by

|ψA (B, k)〉 =L∑j=1

eikj∑mj

(|m1m2 · · ·mL〉 × Tr

[· · ·A[mj−1]B[mj ]A[mj+1] · · ·

]),(3.5)

where B[mj ] is a χ × χ matrix with physical dimension d. Using the shorthand

notation of Eq. (3.2), we denote Eq. (3.5) as

|ψA (B, k)〉 =L∑j=1

eikjj

|[A · · ·ABA · · ·A]〉. (3.6)

where the j on top of the B operator tags its position on the lattice. In the context

of the Single-Mode Approximation, the quasiparticles are usually described by

in terms of a single-site “quasiparticle creation operator” O, such that B[m] =

111

∑m,n

Om,nA[n] which we denote in shorthand as

= or |B〉 = O |A〉 . (3.7)

Note that we could have a quasiparticle tensor B which does not have the form

of Eq. (3.7) (for example O could act on several neighboring sites).∗ However,

for pedagogical reasons, in this chapter we restrict ourselves to B of the form of

Eq. (3.7).

As discussed in Chapter 1, the AKLT chain has an exact low-energy eigenstate

given by

|ψA (B, π)〉 =L∑j=1

(−1)j(S+j )

2 |ψA〉 , (3.8)

where A[m]’s in |ψA〉 are the ground state AKLT MPS tensors of Eq. (2.1) and

B[m] are then given by (following Eq. (3.7))

B[m] =∑

n∈+,0,−(S+)2m,nA

[n], =⇒ B[+] = A[−] = −√

23σ−, B[0] = B[−] = 0.(3.9)

Here B[+] is the only non-trivial matrix, a direct consequence of the (S+)2 operator

acting on spin-1.

3.2.3 Tower of Quasiparticle States

In addition to single quasiparticles, multiple identical quasiparticle states can

be described in the MPS formalism using multiple tensors. For example, the∗B has dχ2 entries while O has d2 entries. Thus, if χ2 > d (dχ2 > d2), not all choices of B

can be expressed in the form of Eq. (3.7).

112

expression for a state with two quasiparticles described by tensor B with momenta

k reads ∣∣ψA (B2, k)⟩≡

(∑j

eikjOj

)2

|ψA〉 . (3.10)

Such a state can also be expressed in the MPS language as

|ψA (B, k)〉 =∑j1 =j2

eik(j1+j2)j1 j2

|[A · · ·ABA · · ·ABA · · ·A]〉+ 2∑j

e2ikjj

|[A · · ·AB2A · · ·A]〉,(3.11)

where we have used the shorthand notation of Eq. (3.2) and defined |Bm〉 ≡

Om |A〉 , m ≥ 1. For example, in the AKLT chain, O = (S+)2 and hence

|B2〉 = 0. Similarly, a state with a number n of B quasiparticles reads

|ψA (Bn, k)〉 =

(∑j

eikjOj

)n

|ψA〉 =∑jl

eik

n∑l=1

jl j1 jl jn

|[A · · ·ABA · · ·ABA · · ·ABA · · ·A]〉,

(3.12)

where B is replaced by Bm if m of the jl’s are equal. If these states |ψA (Bn, k)〉

are eigenstates of the Hamiltonian, they form a tower of (quasiparticle) states

corresponding to the quantum many-body scars.

3.3 Parent Hamiltonian

3.3.1 General Construction

Given an MPS wavefunction |ψA〉 of the form of Eq. (3.4) with a finite bond

dimension χ, we can construct the most general Hamiltonian for which |ψA〉 is a

frustration-free eigenstate.† That is, we can construct a Hamiltonian H that is a†Note that parent Hamiltonian constructions are typically restricted to constructing Hamil-

tonians with |ψA〉 as the ground state. However, such constructions straightforwardly work forhighly excited eigenstates.

113

sum of local terms acting on a finite number of consecutive physical sites such that

each of the local terms vanishes on |ψA〉. Thus, we are looking for Hamiltonians

that satisfy the property

H =L∑j=1

hj, hj |ψA〉 = 0 ∀j, (3.13)

where hj is a local operator with a finite support, j denoting the leftmost site of

this finite support. In general, hj in Eq. (3.13) is a local operator that acts on

several consecutive sites. However, in this work, we always restrict ourselves to

the case where hj is a two-site operator, and the generalization of our formalism

to multisite hj is straightforward. Denoting the two-site hj diagrammatically as

, a sufficient condition for Eq. (3.13) is if the operator hj satisfies

= 0 or hj |AA〉 = 0. (3.14)

To obtain such an operator hj, we consider the MPS on two consecutive sites

|AA〉. |AA〉 can be interpreted as a map from the space of χ× χ matrices Hχ2 to

vectors on the physical Hilbert space of two sites Hd2 as follows:

|AA〉 : Hχ2 −→ Hd2 X 7−→∑m,n

Tr[XA[m]A[n]

]|m,n〉 . (3.15)

Diagrammatically, this map reads X 7−→ . To construct the local

operator hj that satisfies Eq. (3.14), consider the subspace A in the physical

114

Hilbert space of two sites (A ⊆ Hd2) defined as

A ≡ spanX

= spanX

∑m,n

Tr[XA[m]A[n]

]|m,n〉

, (3.16)

where X runs over a complete basis of χ × χ matrices. For example, a complete

basis is the set of matrices X(m,n), each of which has a single non-zero element

given by X(m,n)ij = δm,iδn,j. Such a choice of basis obviously is not unique. For

χ = 2, a convenient choice is 1√21, σ+, σ−, 1√

2σz, which we use in App. C.2.

Since Hχ2 is a χ2-dimensional space, the dimension of A is at most χ2. Provided

d2 > χ2, A has a smaller dimension than the Hd2 and thus A is strictly contained

within (but not equal to) Hd2 . That is, A is a proper subspace of Hd2 (A ⊂ Hd2).

We then can define Ac, the complement of A in Hd2 , as Ac ≡ Hd2/A. For the

local operator hj to vanish on |ψA〉, it is then sufficient to choose any operator

that is supported in Ac. That is the d2 × d2 matrix of hj that satisfies Eq. (3.13)

has the following block-diagonal form in the basis of Ac,A:

hj =

Ac A Z(Ac)j 0

0 0

Ac

A

, (3.17)

where Z(Ac)j is an arbitrary Hermitian matrix with dimension that of Ac. Thus,

the Hamiltonian H of Eq. (3.13) with hj of the form of Eq. (3.17) is a “parent

Hamiltonian” of the the MPS wavefunction |ψA〉. If we also require that |ψA〉 be

the ground state of the Hamiltonian H, we then require Z(Ac)j to be a positive

definite matrix. Note that when Z(Ac)j is not positive definite, |ψA〉 is still an

115

eigenstate of H with area-law entanglement but it is generically located in the

middle of the spectrum. Indeed, it is then an typical example of a quantum scar

of H captured by the Shiraishi-Mori embedding formalism [53].

3.3.2 AKLT State Example

We now illustrate the parent Hamiltonian construction for the AKLT ground state,

with MPS tensors given in Eq. (2.1). Since d2 = 9 > χ2 = 4 for the AKLT MPS,

we are guaranteed that A ⊂ Hd2 , allowing the construction of nearest-neighbor

terms hj that vanish on the MPS state |ψA〉. As shown in App. C.2, the subspace

A defined in Eq. (3.16) can be explicitly computed using the AKLT tensors of

Eq. (2.1). As shown there in Eq. (C.6), we obtain

A = span|J1,1〉 , |J1,0〉 , |J1,−1〉 , |J0,0〉 (3.18)

where |Jj,m〉 is the total angular momentum eigenstate of two spin-1’s with total

spin j and its z-projection m; they are listed in App. C.1. That is, the Hilbert

space of two spin-1’s decomposes into total angular momentum sectors with total

spin 2, 1, or 0 as 1⊗ 1 = 2⊕ 1⊕ 0, and A spans the total spin 1 and 0 subspaces.

In the spin-1/2 Schwinger boson language, this is evident as there is a spin 1/2

singlet between any two adjacent sites. The remaining spin 1/2’s, one on each

adjacent site, cannot clearly sum to spin-2. Hence its orthogonal subspace Ac

spans the total spin-2 subspace, i.e.

Ac = span|J2,2〉 , |J2,1〉 , |J2,0〉 , |J2,−1〉 , |J2,−2〉. (3.19)

116

Thus, following Eq. (3.17), with the elements of Z(Ac)j defined as (Z

(Ac)j )m,n =

z(m,n)j , the most general nearest-neighbor Hamiltonian with |ψA〉 as a frustration-

free eigenstate reads

H =∑j

hj, hj =∑m,n

z(m,n)j |J2,m〉 〈J2,n| (3.20)

with Hermiticity imposing z(n,m)j = (z

(m,n)j )∗. However, imposing symmetries on

the Hamiltonians restricts the form of Zj. For example, translation invariance

requires that Zj be independent of j. Sz-spin conservation U(1) symmetry requires

that Zj be diagonal, since the operators |J2,m〉 〈J2,n| do not preserve the spin Sz

for m 6= n. Furthermore, imposing SU(2) symmetry on the parent Hamiltonian

requires that all the operators |J2,m〉 〈J2,m| appear with the same coefficient in the

Hamiltonian. Thus, with translation invariance and SU(2) symmetry, the local

term of the parent Hamiltonian is uniquely determined to be

hj = c2∑

m=−2

|J2,m〉 〈J2,m| = c P (2,1), (3.21)

where c is an arbitrary constant P (2,1) is the projector of two spin-1’s onto total

spin 2, which is nothing but the local term of the AKLT Hamiltonian.

3.4 Quantum Scarred Hamiltonians

Having constructed the most general nearest-neighbor Hamiltonian for which |ψA〉

is a frustration-free eigenstate, we would like to determine the set of conditions

on Z(Ac)j in Eq. (3.17) such that the Hamiltonian H exhibits a quasiparticle tower

of states. For example, in the case of the AKLT MPS, we know that the AKLT

117

Hamiltonian (Z(Ac)j = 1) exhibits a tower of states [55, 56]. Here we show that

there are other choices of Z(Ac)j for which the states in the AKLT tower are eigen-

states.

3.4.1 One Quasiparticle Eigenstate

We now illustrate the formalism to construct a Hamiltonian with a single quasi-

particle eigenstate in addition to a frustration free MPS eigenstate, similar to the

case discussed in Sec. 3.2.2. That is, given an MPS wavefunction |ψA〉, we want

to obtain a Hamiltonian, with |ψA〉 as an eigenstate, that also has a quasiparti-

cle eigenstate of the form |ψA (B, k)〉 of Eq. (3.5) with energy E . As we show in

App. C.3, a sufficient local condition is (using the shorthand notation of Eq. (3.3))

= E or hj(|BA〉+ eik |AB〉

)= E

(|BA〉+ eik |AB〉

),

(3.22)

where

≡ + eik (3.23)

To find operators that satisfy Eq. (3.22), similar to Eq. (3.15), we view(|BA〉+ eik |AB〉

)as a map from the space of χ × χ matrices Hχ2 to the physical Hilbert space of

two sites Hd2 :

(|BA〉+ eik |AB〉

): Hχ2 −→ Hd2 , X 7−→ (3.24)

118

We define the subspace B (⊆ Hd2) as

B ≡ spanX

= spanX

∑m,n

Tr[X(B[m]A[n] + eikA[m]B[n]

)]|m,n〉

,

(3.25)

where X runs over a complete basis of χ × χ matrices. In terms of a single-site

quasiparticle creation operator O for which the tensors B and A satisfy Eq. (3.7),

the subspace B reads

B =(O ⊗ 1+ eik1⊗ O

)︸ ︷︷ ︸

O

A ≡ spanO |ψ〉 : |ψ〉 ∈ A. (3.26)

Since B has a dimension of at most χ2, if d2 > χ2, B is a proper subspace of Hd2

(B ⊂ Hd2). Defining the complement as Bc ≡ Hd2/B, the hj satisfying Eq. (3.22)

reads

hj =

Bc B Z(Bc)j 0

0 E1

Bc

B

, (3.27)

where Z(B)j is an arbitrary matrix with the same dimension as Bc. However, to

obtain hj that satisfies both Eq. (3.14) and Eq. (3.22) with E 6= 0, it is essential

that A lies within the subspace Bc in Eq. (3.27). In other words, we require

A ⊆ Bc or B ⊆ Ac. In fact, operators O and momentum eik for which B satisfies

this condition can be found by ensuring the orthogonality of states in A and B by

119

solving the linear equation

= 0 ⇐⇒ = −eik . (3.28)

The term hj then has the structure

hj =

Ac/B B AZ

(Ac/B)j 0 0

0 E1 0

0 0 0

Ac/B

B

A

, (3.29)

where Z(Ac/B)j here is an arbitrary Hermitian matrix with the same dimension as

Ac/B.

3.4.2 Tower of Quasiparticle Eigenstates

Given the most general Hamiltonian of the form of Eq. (3.29) that has a single-

quasiparticle eigenstate, we now wish to construct a Hamiltonian with a tower of

quasiparticle eigenstates of the form of Eq. (3.12) in Sec. 3.2.3. One way to do

so is to impose emergent constraints on the quasiparticles, similar to the tower of

states in the AKLT chain [55] (as we show in Sec. 3.4.3) and the ones in Ref. [76].

For example if the quasiparticles are naturally constrained to be at least one site

away from each other, the quasiparticles do not interact with each other under a

nearest-neighbor Hamiltonian and, as we show in this section, we can construct

eigenstates composed of multiple identical quasiparticles. In terms of the MPS,

120

such a condition reads

|BB〉 = = 0 ⇐⇒(O ⊗ O

)A = 0, (3.30)

and O |B〉 = O2 |A〉 = 0, (3.31)

A is defined in Eq. (3.16). Eq. (3.30) prohibits two quasiparticles on neighboring

sites, and Eq. (3.31) prohibits two quasiparticles on the same site. As shown in

App. C.3, these conditions result in a tower of exact eigenstates of the Hamiltonian

H with local terms of the form of Eq. (3.29). The states of the tower have the

form |S2n〉

|S2n〉 = Pn |ψA〉 , H |S2n〉 = 2En |S2n〉 , n ≤ L/2, P =L∑j=1

eikjOj. (3.32)

Due to Eq. (3.30), the tower is guaranteed to end on the state after (L/2 + 1)

applications of P on the state |ψA〉 since

PL2 |ψA〉 ∝ (|BABA · · · 〉 ± |ABABA · · · 〉) =⇒ P

L2+1 |ψA〉 = 0. (3.33)

3.4.3 Models with AKLT Tower of States

We now show that the scars in the AKLT chain can be explained in this formalism,

and we construct a family of nearest-neighbor Hamiltonians for which all the scars

of the AKLT chain are eigenstates. We start with the spin-1 AKLT ground state

MPS of Eq. (2.1), and take the operator O and momentum k to be

O = (S+)2, k = π. (3.34)

121

The subspace B defined in Eq. (3.26) then reads

B =((S+)2 ⊗ 1− 1⊗ (S+)2

)︸ ︷︷ ︸≡(S+)2

A, (3.35)

where A for the AKLT MPS is shown in Eq. (3.16). We can compute the subspace

B by noting two important properties of the operator (S+)2: (i) it is a spin-2

operator, (ii) it is antisymmetric (i.e. (S+)2 −→ −(S+)2) under exchange of the

two sites involved in Eq. (3.35). We then can deduce the following (see Sec. C.2

in Appendix C for a derivation):

1. The vector |J1,1〉 in A with spin 1 vanishes under the action of (S+)2 since

a vector with spin 3 cannot be formed from two spin-1’s.

2. Since the vector |J0,0〉 in A is symmetric under exchange, it vanishes under

the action of (S+)2 since an antisymmetric vector with spin 2 cannot be

formed from two spin-1’s.

3. The remaining vectors in A: |J1,0〉 and |J1,−1〉 are antisymmetric under

exchange, and thus under the action of (S+)2 result in symmetric states

with spins 2 and 1 respectively (i.e. |J2,2〉 and |J2,1〉 respectively).

Thus,

A = span|J0,0〉 , |J1,−1〉 , |J1,0〉 , |J1,1〉 =⇒ B = span|J2,1〉 , |J2,2〉,

=⇒ Ac/B = span|J2,0〉 , |J2,−1〉 , |J2,−2〉. (3.36)

Clearly, A and B in Eq. (3.36) are orthogonal subspaces, satisfying the required

condition. Furthermore, Eq. (3.30) is satisfied since ((S+)2 ⊗ (S+)2), by virtue of

122

being a spin-4 operator, vanishes on all states of A. Eq. (3.31) is also satisfied

since ((S+)2)2= 0. Thus, the most general nearest-neighbor Hamiltonian with

the AKLT tower of states as eigenstates has the form of Eq. (3.29). It reads

H =∑j

hj, hj = E (|J2,1〉 〈J2,1|+ |J2,2〉 〈J2,2|)+0∑

m,n=−2

z(m,n)j |J2,m〉 〈J2,n|. (3.37)

with z(n,m)j = (z

(m,n)j )∗. Note that E in Eq. (3.37)) is only an overall scale. This 6-

parameter family of nearest-neighbor Hamiltonians was also obtained very recently

in Ref. [61]. If we demand conservation of Sz, we need to set z(m,n)j ∝ δm,n, yielding

a three-dimensional family of Hamiltonians. The AKLT Hamiltonian is recovered

by setting z(m,n) = Eδm,n.

Instead of assuming O and k in Eq. (3.34), we can also arrive at that choice by

brute force solving Eq. (3.28) for eik and O given the AKLT MPS of Eq. (2.1).

Solving for the 10 variables (k and 9 parameters in O) using symbolic computation

software, we obtain the solutions

k = π and O ∈ span(S+)2, S+, Sz, 2(Sz)2 − S+, S−, S−, Sz, (S−)2

(3.38)

where ·, · represents the anticommutator. This guarantees the existence of

Hamiltonians with local terms of the form Eq. (3.29) having one-quasiparticle

eigenstates with energy E of the form

|ψA (B, π)〉 =∑j

(−1)jOj |ψA〉 . (3.39)

where Oj is chosen from Eq. (3.38). In general, the subspace B in Eq. (3.29)

123

depends on the choice of O in Eq. (3.38), and generically we obtain distinct fam-

ilies of Hamiltonians for distinct O’s. The families of Hamiltonians obtained for

different choices of O intersect at the AKLT Hamiltonian (i.e. when Zj = E1 in

Eq. (3.29)). The five independent excited states there span the entire multiplet

of spin-2 magnon state, as shown in App. C.4.‡

We further impose Eqs. (3.30) and (3.31) on O, and by numerical brute force

we obtain precisely two choices for O:

O = (S+)2 or O = (S−)2. (3.40)

These are the only choices of single-site operators that generate the tower of

states starting from the AKLT MPS. The fact that the AKLT state satisfies

Eq. (3.30) is a consequence of “string order” in the AKLT ground state [165,173–

175]. That is, when decomposed in the spin-1 product state basis, the AKLT

ground state has zero weight on configurations that have the form |· · ·+ 0n + · · · 〉

or |· · · − 0n − · · · 〉 for n ≥ 0, where 0n represents a string of n 0’s. In particular,

nearest neighbor configurations of |++〉 and |−−〉 do not appear in the AKLT

ground state. Since the operators (S+)2 ⊗ (S+)2 and (S−)2 ⊗ (S−)2) are non-

vanishing only on the configurations |−−〉 and |++〉 respectively, they vanish on

the AKLT ground state, thus satisfying Eq. (3.30). These two towers are actually

equivalent in the AKLT chain since they correspond to highest and lowest states

of the same multiplet of the SU(2) symmetry. However, since the B subspaces

depend on the choice of O in Eq. (3.40), we can deform away from the AKLT

Hamiltonian (by breaking the SU(2) symmetry) and preserve only one tower of‡The AKLT Hamiltonian is SU(2) symmetric, and hence an eigenstate with spin s is (2s+1)-

fold degenerate, e.g. the spin-2 magnon state is 5-fold degenerate.

124

states: either the highest weight state or the lowest weight states of the SU(2)

multiplet of the AKLT tower of states.

3.5 New Families of AKLT-like Quantum Scarred Hamil-

tonians

3.5.1 Quantum Scarred Hamiltonians from Generalized AKLT MPS

Having established the formalism to construct quantum scarred models starting

from an MPS, we now deform away from the AKLT MPS and obtain new families

of quantum scarred Hamiltonians. Since we start with a different MPS, the tower

of states presented in this section is distinct from the AKLT tower of states. In

particular, we consider the following generalization of the AKLT MPS of Eq. (2.1)

A[+] = c+σ+, A[0] = c0σ

z, A[−] = c−σ−, (3.41)

where one of c+, c0 and c− is fixed by the normalization of the MPS wavefunction

|ψA〉. Note that for (c+, c0, c−) = 1√α+1

(√α,−1,−

√α), the MPS of Eq. (3.41)

coincides with some of the ones considered in Ref. [176]. By numerical brute

force, we find that Eqs. (3.28), (3.30) and (3.31) are satisfied for for the MPS A

if O = (S+)2 and k = π. Thus, there exist Hamiltonians for which the MPS of

Eq. (3.41) are frustration-free eigenstates and a tower of eigenstates can be built

from them with the same raising operator P as that of the AKLT tower of states.

125

As we show in App. C.2, the subspaces A and B for the MPS of Eq. (3.41) read

A = span|K0,0〉 , |K1,−1〉 , |K1,0〉 , |K1,1〉

=⇒ B = span|K2,1〉 , |K2,2〉

=⇒ Ac/B = span|K2,0〉 , |K2,−1〉 , |K2,−2〉, (3.42)

where we have defined

|K0,0〉 ≡c+c− (|+−〉+ |−+〉) + 2c20 |00〉√

2 (|c+c−|2 + 2|c0|4)

|K2,0〉 ≡−c20 (|+−〉+ |−+〉) + c+c− |00〉√

|c+c−|2 + 2|c0|4

|Kj,m〉 ≡ |Jj,m〉 if (j,m) /∈ (0, 0), (2, 0). (3.43)

Note that the subspaces A and B in Eq. (3.42) are orthogonal irrespective of

the values of (c+, c0, c−). Furthermore Eqs. (3.30) and (3.31) are satisfied for the

same reasons as those for the AKLT MPS (see Sec. 3.4.3). Since the dimensions

of the subspaces A and B are the same as that in the AKLT case (see Eq. (3.36)),

we can similarly derive the 6 parameter family of Hermitian nearest-neighbor

Hamiltonian that has a tower of states generated from the MPS eigenstate of

Eq. (3.41), composed of the local term hj of the form of Eq. (3.17). Thus, the

most general Hamiltonian with such a tower reads

H =∑j

hj, hj = E (|K2,1〉 〈K2,1|+ |K2,2〉 〈K2,2|) +0∑

m,n=−2

z(m,n)j |K2,m〉 〈K2,n|.

(3.44)

126

3.5.2 Deformation to Integrability

Continuous deformations maintaining exact ground states connect the AKLT and

the spin-1 biquadratic chains [153]. Here we give a continuous deformation that

preserves a tower of exact quasiparticle states as well. The biquadratic chain

is integrable [177–179], in contrast to the other Hamiltonians considered in this

paper. We start with the observation that

(S · S

)2= (|+−〉+ |−+〉 − |00〉) (〈+−|+ 〈−+| − 〈00|) + 1, (3.45)

where S · S is the usual two-site Heisenberg interaction:

S · S ≡ S+ ⊗ S− + S− ⊗ S+ + Sz ⊗ Sz. (3.46)

Thus, if we set c20 = c+c−, using Eq. (3.43), Eq. (3.45) can be written as

1

3

((S · S

)2− 1

)= |K2,0〉 〈K2,0| . (3.47)

The Hamiltonian built from Eq. (3.47) is thus of the form of Eq. (3.44) with the

parameters E = 0 and z(m,n)j = δm,0δn,0. Thus, in the space of quantum scarred

Hamiltonians considered here, the AKLT and pure biquadratic models are located

at the following points

AKLT :

(E , z(m,n)j ,

c+c0,c−c0

)=

(1, δm,n,−

√2,√2)

Biquadratic :

(E , z(m,n)j ,

c+c0,c−c0

)= (0, δm,0δn,0, 1, 1) .

127

We consider a path between the two given by

(E , z(m,n)j , c+c0, c−c0) = (2 cos θ , 2 cos θ δm,n + (csc θ + 2 sin θ − 4 cos θ)δm,0δn,0,

−√cot θ − 1,

√cot θ − 1). (3.48)

We recover the AKLT chain (up to a constant factor and constant shift) by setting

θ = cot−1 3, and the pure biquadratic model by setting θ = π/2. The Hamiltonian

parametrized by θ reads

H =∑j

[cos θ

(Sj · Sj+1

)+ sin θ

(Sj · Sj+1

)2+ cos θ (cot θ − 3) P 00

j,j+1

], (3.49)

up to an overall factor and constant shift. We thus recover the usual bilinear-

biquadratic chain plus an additional term proportional to P 00 = |00〉 〈00|. The

model of Eq. (3.49) thus has a tower of exact eigenstates with spacing E = 4 cos θ

starting from the ground state.

We note that at the purely biquadratic point (θ = π2), all the states of the tower

are degenerate (E = 0). Indeed, we can verify that

[(S+

j )2 − (S+

j+1)2,(Sj · Sj+1

)2]= 0 =⇒

[∑j

(−1)j(S+j )

2,∑j

(Sj · Sj+1

)2]= 0.

(3.50)

The operator generating the tower of states is thus a symmetry of this integrable

Hamiltonian.

128

3.6 New Type of Quantum Scars: Two-Site Quasiparti-

cle Operators

In the previous sections, we assumed that the quasiparticle that constitutes the

tower is a one-site operator, and we found a family of quantum scarred Hamil-

tonians where the quasiparticle creation operator is the same as the one in the

AKLT chain. Here we relax the constraint that the quasiparticle be a single-site

operator, and find examples of Hamiltonians that contain a tower of states of a

different type. In this way, we clearly show that our construction gives rise to

many Hamiltonians which contain scar states.

3.6.1 Scars with two-site quasiparticles

We first set up the general formalism for obtaining Hamiltonians that have a

two-site quasiparticle tower of states. Similarly to the case of single-site quasi-

particles, we focus on Hamiltonians that satisfy Eq. (3.13), i.e. those which have

a frustration-free MPS eigenstate |ψA〉 of the form Eq. (3.4). As illustrated in

Sec. 3.3, the most general nearest-neighbor Hamiltonian with such a property has

a local term of the form of Eq. (3.17). A two-site quasiparticle BB has the form

BB[m,n]

=∑l,r

O(2)ml,nrA

[l]A[r], which in shorthand, we write

= or∣∣∣BB⟩ = O(2) |AA〉 , (3.51)

129

where O(2) is a nearest-neighbor two-site operator. The wavefunction with the

quasiparticle dispersing with momentum k then reads

∣∣∣ψA(BB, k)⟩ =∑j

eikjj,j+1∣∣∣[A · · ·ABBA · · ·A]⟩. (3.52)

We can show that a set of sufficient conditions for the existence of an eigenstate

of the form of Eq. (3.52) with energy (2E1 + E2) reads

= 0 or hj |AA〉 = 0, (3.53)

= E2 or (hj − E2)∣∣∣BB⟩ = 0, (3.54)

+ eik = E1(

+ eik)

or

(hj+1 − E1)j+1∣∣∣BBA⟩+ eik(hj − E1)

j+1∣∣∣ABB⟩ = 0 (3.55)

where we have used the shorthand notations of the form of Eq. (3.3). We now

proceed to construct local terms hj that satisfy Eqs. (3.53)-(3.55). Hamiltonian

terms satisfying Eqs. (3.53) and (3.54) can be built similarly to the single-site

quasiparticle case. That is, we first construct the subspaces A and B, where A is

defined in Eq. (3.16) and B is now defined as

B ≡ spanX

= spanX

∑m,n

Tr[BB[m,n]

] |m,n〉 = O(2)A. (3.56)

130

The Hamiltonian term that satisfies Eqs. (3.53) and (3.54) then reads

hj =

Ac/B B AZ

(Ac/B)j 0 0

0 E21 0

0 0 0

Ac/B

B

A

, (3.57)

where Z(Ac/B)j here is an arbitrary Hermitian matrix with the same dimension as

Ac/B, Ac being the complement of A.

To aid in finding a solution to Eq. (3.55), we decompose the two-site quasipar-

ticle BB as

= or∣∣∣BB⟩ =

∣∣∣BlBr

⟩, (3.58)

where Bl and Br are one-site MPSs. Note that the decomposition of Eq. (3.58)

is not unique, and B[α]l and B

[α]r (where α is the physical index) need not be

square matrices. That is, the contracted auxiliary index in Eq. (3.58) (denoted

by a dashed line) can have a different dimension than that of contracted auxiliary

indices (denoted by solid lines). As we show below, it is sufficient to find a two-site

operator hj and a one-site MPS C such that

= E1 + or(hj − E1

) j∣∣∣BrA⟩=

j∣∣∣CBr

⟩, (3.59)

= E1 − e−ik or(hj − E1

) j∣∣∣ABl

⟩= −e−ik

j∣∣∣BlC⟩.(3.60)

131

By left-multiplying Eq. (3.59) by∣∣∣Br

⟩and replacing j by j + 1, we obtain

(hj+1 − E1

) j+1∣∣∣BlBrA⟩=(hj+1 − E1

) j+1∣∣∣BBA⟩ =

j+1∣∣∣BlCBr

⟩, (3.61)

where we have used Eq. (3.58). Similarly, by right-multiplying Eq. (3.60) by∣∣∣Bl

⟩,

we obtain

(hj − E1

) j+1∣∣∣ABlBr

⟩=(hj − E1

) j+1∣∣∣ABB⟩ = −e−ikj+1∣∣∣BlCBr

⟩, (3.62)

Using Eqs. (3.61) and (3.62), we immediately see that Eq. (3.55) follows from

Eqs. (3.59) and (3.60). Generically it is not clear we can find terms hj of the form

of Eq. (3.57) (for some Z(Ac/B)) that also satisfy Eqs. (3.59) and (3.60). However,

in the next subsection, we show that when we restrict ourselves to a particular

form of the MPS matrices, we can fix the form of Z(Ac/B) such that Eq. (3.55) is

satisfied.

Similar to the one-site quasiparticle, we can obtain a tower of equally spaced

eigenstates composed of the BB quasiparticles if BB obeys the additional con-

straints that generalize Eq. (3.31). A sufficient condition is to constrain the quasi-

particles to be at least one site away from each other. That is, we require

O(2)j

∣∣∣BB⟩ = 0, (3.63)

O(2)j

j+1∣∣∣ABB⟩ = O(2)j+1

j+1∣∣∣BBA⟩ = 0, (3.64)∣∣∣BBBB⟩ = = 0,

(3.65)

132

where O(2)j is the two-site quasiparticle creation operator. Similar to Eq. (3.32),

we then obtain a tower of equally spaced eigenstates ∣∣∣S(2)

3n

⟩, where

∣∣∣S(2)3n

⟩= (P(2))n |ψA〉 , H

∣∣∣S(2)3n

⟩= n(2E1 + E2)

∣∣∣S(2)3n

⟩, n ≤ L/3,

P(2) =L∑j=1

eikjO(2)j (3.66)

Note that the tower is guaranteed to end on the state after (L/3 + 1) applications

of P(2) on the state |ψA〉 since we can have at most L/3 BB quasiparticles on the

chain that satisfy the constraints of Eqs. (3.63)-(3.65).

3.6.2 Concrete example: Perturbed Potts MPS

We now provide a concrete example of a model where we find a two-site quasipar-

ticle tower of states of the form discussed in Sec. 3.6.1. Throughout this section,

we use as an example the MPS

A[+] =1√2σ+, A[0] =

1√212×2, A[−] =

1√2σ−. (3.67)

This MPS is the ground state of the Hamiltonian [153, 171]

HPP =∑j

(S+j2S−j+1

2 − S+j S

−j+1 + h.c.

). (3.68)

As apparent, HPP has a U(1) symmetry corresponding to the total spin∑

j Szj . In

addition, it has a spin-flip symmetry (Pz) given by S+j ↔ S−

j and Szj ↔ −Szj , and

inversion symmetry (I), defined by taking all operators at site j to site L+ 1− j

for a chain of length L. The Hamiltonian of Eq. (3.68) arises from perturbing the

133

S3-invariant three-state Potts chain by shortest-range U(1)-invariant interaction

[171].

We show that HPP has a two-site quasiparticle tower of exact eigenstates, and

derive a family of Hamiltonians with a similar tower of states. In order to do so,

we construct the subspace A defined in Eq. (3.16). As shown in App. C.5, the

subspace reads

A = span|J2,0〉 , |J2,−1〉 , |J1,0〉 , |J2,1〉. (3.69)

We now consider the quasiparticle creation operator

P(2) =∑j

(−1)j((S+

j )2S+

j+1 + S+j (S

+j+1)

2)︸ ︷︷ ︸

=O(2)j

. (3.70)

As shown in Eq. (C.32) in App. C.5, the subspace B defined in Eq. (3.56) reads B =

|J2,2〉. Thus, a nearest-neighbor Hamiltonian term hj that satisfies Eqs. (3.53)

and (3.54) has the form of Eq. (3.57).

We now obtain a general solution to Eqs. (3.59) and (3.60) using the MPS of

Eq. (C.28) and hj of the form of Eq. (3.57), where the subspaces A and B are

obtained in Eqs. (C.30) and (C.32) respectively. In particular, we use the following

properties of hj:

hjB = E2B =⇒ hj |++〉 = E2 |++〉 , (3.71)

hjA = 0 =⇒ hj1√2(|+0〉+ |0+〉) = 0. (3.72)

It is convenient to represent MPS tensors as vectors on the physical indices with

matrix coefficients. For example, we express the MPS A of Eq. (3.67) (with

134

coefficients of Eq. (3.87) for generality) as

|A〉 = c+σ+ |+〉 + c0σ

0 |0〉 + c−σ− |−〉 =

∑α∈+,0,−

cασα |α〉. (3.73)

Consequently multisite MPS can be obtained with a matrix multiplication of the

coefficients and a tensor product over the physical indices. Using the operator

O(2) = (S+)2 ⊗ S+ + S+ ⊗ (S+)2, using Eq. (3.73), we straightforwardly obtain

the expression for the BB quasiparticle tensor defined in Eq. (3.51):

∣∣∣BB⟩ = 2c0c−σ− |++〉 = 2c0c−

0 0

1 0

|++〉 , (3.74)

where the 2×2 matrix is over the auxiliary indices and the |·〉 is over the physical

index of the BB tensor. As shown in Eq. (3.58), we can always decompose (in a

non-unique way)∣∣∣BB⟩ =

∣∣∣BlBr

⟩= |Bl〉 ⊗ |Br〉. Applied to Eq. (3.74), we get

∣∣∣Bl

⟩=√

2c0c−

0

1

|+〉 , ∣∣∣Br

⟩=√

2c0c−

(1 0

)|+〉 . (3.75)

Using Eqs. (3.73) and (3.75),∣∣∣ABl

⟩and

∣∣∣BrA⟩

read

∣∣∣ABl

⟩=√2c0c−

c+

0

|++〉+

0

c0

|0+〉 ,

∣∣∣BrA⟩=√2c0c−

((0 c+

)|++〉+

(c0 0

)|+0〉

). (3.76)

135

The most general |C〉 in this case has the form |C〉 =∑

α∈+,0,−C [α] |α〉, where

C [α] are numbers. Consequently, using Eq. (3.75),∣∣∣BlC

⟩and

∣∣∣CBr

⟩read

∣∣∣BlC⟩=√

2c0c−∑

α∈+,0,−

0

C [α]

|+α〉, ∣∣∣CBr

⟩=√

2c0c−∑

α∈+,0,−

(C [α] 0

)|α+〉.

(3.77)

Using Eqs. (3.76) and (3.77), Eq. (3.59) reads

c+0

(hj − E1) |++〉+

0

c0

(hj − E1) |0+〉 = ∑α∈+,0,−

0

C [α]

|+α〉=⇒

c+0

(E2 − E1) |++〉+

0

c0

(hj − E1) |0+〉 = ∑α∈+,0,−

0

C [α]

|+α〉,(3.78)

where we have used Eq. (3.71). Equating the two components of the row vector

in Eq. (3.78), we obtain

c+ (E2 − E1) |++〉 = 0 =⇒ E2 = E1 ≡ E ,(hj − E1

)|0+〉 =

∑α∈+,0,−

C [α]

c0|+α〉.

(3.79)

Similarly, solving Eq. (3.60), we obtain

(hj − E

)|0+〉 =

∑α∈+,0,−

C [α]

c0|+α〉, (3.80)

136

where E ≡ E1 = E2. Adding Eqs. (3.79) and (3.80), we obtain

(hj − E

)(|+0⟩+|0+⟩)√

2=

∑α∈+,0,−

C[α]

c0

(|+α⟩+|α+⟩)√2

=⇒ hj |J2,1〉 =(E + C[0]

c0

)|J2,1〉+

√2C

[+]

c0|J2,2〉+ C[−]

c0|J2,0〉 .

(3.81)

However, using Eq. (3.72), we obtain C [α] = −c0Eδα,0. Further, subtracting

Eqs. (3.79) and (3.80), we obtain

(hj − E

)|J1,1〉 = E |J1,1〉 =⇒ hj |J1,1〉 = 2E |J1,1〉 . (3.82)

Thus, Eqs. (3.59) and (3.60) are satisfied if

E1 = E2 ≡ E , C [α] = −c0Eδα,0, hj |J1,1〉 = 2E |J1,1〉 . (3.83)

The Hamiltonian term that satisfies Eqs. (3.53)-(3.55) then reads

hj =

(Ac/B)/C C B A

Z(Ac/B)/Cj 0 0 0

0 2E1 0 0

0 0 E1 0

0 0 0 0

(Ac/B)/C

C

B

A

, (3.84)

where Z(Ac/B)/Cj is an arbitrary matrix with the same dimension as (Ac/B)/C and

we have defined the subspace C as C = |J1,1〉.

Any Hamiltonian of the form of Eq. (3.84) hosts a quasiparticle excited state

137

of the form of Eq. (3.52) created by the operator P(2) of Eq. (3.70). In addi-

tion, by brute force we have verified that the operator O(2) in Eq. (3.70) satisfies

Eqs. (3.63)-(3.65) with the MPS of Eq. (3.67). Thus we find a 6-parameter family

of Hamiltonians hosting a tower of two-site quasiparticles with energies E = 3nE .

The Hamiltonians are of the form H =∑

h hj, where

hj = E |J2,2〉 〈J2,2|+ 2E |J1,1〉 〈J1,1|+2∑

m,n=0

z(m,n)j |Jm,−m〉 〈Jn,−n| . (3.85)

As before, E in Eq. (3.85) is merely an overall scale. Indeed, the terms for the

perturbed Potts model of Eq. (3.68) read (up to overall constant factors and

energy shifts)

hj = |J2,2〉 〈J2,2|+ |J2,−2〉 〈J2,−2|+ 3 |J0,0〉 〈J0,0|+ 2 (|J1,1〉 〈J1,1|+ |J1,−1〉 〈J1,−1|) ,

(3.86)

which can be obtained from Eq. (3.85) by setting E = 1 and z(m,n)j = (3−m)δm,n.

We can indeed repeat this exercise for the generalized perturbed Potts MPS

that reads

A[+] = c+σ+, A[0] = c012×2, A[−] = c−σ

−, (3.87)

where one of c+, c0, and c− is fixed by normalization. We note that we can build a

scar-preserving deformation from the perturbed Potts model of Eq. (3.68) to the

spin-1 pure biquadratic Hamiltonian similar to the scar-preserving deformation of

the AKLT chain illustrated in Sec. 3.5.2. However, unlike in the AKLT case, the

quasiparticle operator P(2) of Eq. (3.70) is not a symmetry of the pure biquadratic

Hamiltonian.

138

3.7 Conclusions

We have provided a formalism to search and construct quantum scarred models

starting from a Matrix Product State wavefunction. The scarred Hamiltonians we

construct have a quasiparticle tower of exact eigenstates in their spectra. We have

illustrated our method thoroughly for single-site quasiparticles by constructing a

6-parameter family of nearest-neighbor Hamiltonians that have the exact quantum

scars of the AKLT chain as eigenstates. Applying our construction to a more

general class of MPS wavefunctions, we showed that the scars of AKLT chain [55,

56] can be continuously deformed to a symmetry of the pure biquadratic spin-1

model, an integrable model. Further, we generalized our construction to the case

of two-site quasiparticles and we obtain new types of quantum scarred models.

We illustrated these results with the help of a concrete example of the perturbed

Potts model [171], which we show that hosts a tower of exact eigenstates composed

of two-site quasiparticles.

We believe that our formalism can be generalized to include a wide variety of

known models with quantum scars, including the spin-S AKLT chains. We also

expect that many more models with quantum scars can be obtained by relaxing

several assumptions introduced for pedagogical reasons in this work, such as peri-

odic boundary conditions, single-site or two-site quasiparticle creation operators

or nearest-neighbor Hamiltonians. It would also be interesting to work out the

exact relation between the MPS construction of scars and the unified formalisms

proposed in Refs. [61] and [62], and formulate a dimension independent under-

standing of scars. It should also be possible to extend our formalism to higher

dimensions using Projected Entangled Pair States (PEPS) [180] and search for

139

higher dimensional quantum scarred models, a question we defer for future work.

On a different note, given that the PXP model has exact MPS eigenstates [75,181]

as well as an approximate MPS ground state [182, 183], it is natural to ask if the

scars exhibited there have any connections to the formalism developed here. Fur-

thermore, the deformation to integrability raises questions of whether quantum

scarred Hamiltonians are always connected to integrable ones, as suggested by

numerical explorations around the PXP model [184].

140

4Quantum Scars from Spectrum

Generating Algebras

4.1 Introduction

In this chapter, we focus on the analytically tractable quantum scars with equally

spaced towers of states in more physically relevant electronic systems rather than

hard-core bosonic spin models which are hard to naturally realize in experiments.

141

Equally spaced towers of states have been known to occur in the celebrated Hub-

bard models since the seminal work of Yang that introduced the mechanism of

η-pairing [185]. The existence of η-pairing and the related Off-Diagonal Long-

Range Order (ODLRO) is attributed to the understanding of a pseudospin SU(2)

symmetry of the Hubbard model [186, 187]. There has since been a vast amount

of literature studying the existence and properties of η-pairing and its generaliza-

tions to a wide range of models [188–197]. Despite this large body of literature,

the natural connection between the η-pairing states and the infinite-temperature

quantum dynamics of the Hubbard models has not been extensively explored apart

from the one-dimensional case, where the Hubbard model is fully integrable [198].

Notable exceptions include Ref. [152], which computed the entanglement of some

analytically tractable eigenstates [187] of the D-dimensional Hubbard models, and

Ref. [199], where the effect of η-pairing on many-body localized Hubbard models

was numerically explored. However, the analytically tractable η-pairing states in

the D-dimensional Hubbard models [187] are not examples of quantum scars even

though some of them have low entanglement since it was proven that they are the

only eigenstates in their respective quantum number sectors [152]. That is, they

do not appear to be mixed with ETH-satisfying states in the spectrum with the

same set of quantum numbers.

Given the similarity with quantum scars, it is natural to explore the precise

connection of these η-pairing towers of states and quantum many-body scars.

In particular, we ask if it is possible to deform the Hubbard model such that

the pseudospin - η- symmetry is broken (and hence most of the η-pairing eigen-

states would cease to be eigenstates) while preserving a subset of the analytically

tractable eigenstates of the Hubbard model, which would then become examples

142

of quantum many-body scars. To do so, we first recast η-pairing as a real-space

phenomenon in contrast to the momentum-space approach employed in most of

the literature. This makes clear the minimal conditions necessary for the existence

of η-pairing, and unravels a large class of Hubbard models with disorder and/or

spin-orbit coupling that exhibit η-pairing. We refer to this algebraic structure as

a Spectrum Generating Algebra (SGA). We then introduce the concept of a Re-

stricted Spectrum Generating Algebra (RSGA), and we show that perturbations

can be added to the Hubbard models that enable some of the analytically tractable

η-pairing towers of the Hubbard models to survive as eigenstates of the perturbed

models. We show analytically that these states, which have a low entanglement

entropy [152], lie in the bulk of the spectrum of their quantum number sectors, and

thus form examples of quantum many-body scars. We show that these RSGAs

also appear in existing models of quantum scars in the literature, for example, the

AKLT model [55] and the spin-1 XY model [57]. We note that related algebraic

structures have appeared in the literature in the past in the context of Generalized

Hubbard Models [191], and more recently in the context of unifying formalisms

for quantum scarred models [61, 62].

This chapter is organized as follows. In Sec. 4.2 we review the Fermi-Hubbard

model and the existence of a Spectrum Generating Algebra (SGA), i.e. the η-

pairing states. In Sec. 4.3, we illustrate the generalization of the η-pairing states

to Hubbard models on arbitrary graphs with disorder in the hopping terms and

with spin-orbit coupling. We discuss some examples in Sec. 4.4. In Sec. 4.5, we

introduce the concept of Restricted Spectrum Generating Algebra (RSGA), which

captures the behavior of several known quantum scarred models, and we intro-

duce perturbations to the (generalized) Hubbard models that realize an RSGA.

143

There, we analytically show that the tower of eigenstates realized by the RSGA

are quantum many-body scars of the perturbed Hamiltonians by deriving the con-

ditions for which the states are in the bulk of the spectra of their quantum number

sectors. In Sec. 4.8, we comment on connections between the RSGA formalism

and quantum scarred models in the literature. We conclude with a discussion of

future directions in Sec. 4.9.

4.2 Review of η-pairing in the Hubbard Model

We review the construction of η-pairing states in the Fermi-Hubbard model (that

we also refer to as the “Hubbard model”), first obtained in Refs. [185–187]. The

Hubbard Hamiltonian is given by

HHub =∑

σ∈↑,↓

−t∑⟨r,r′⟩

(c†r,σcr′,σ + h.c

)− µ

∑r

c†r,σcr,σ

+ U∑r

nr,↑nr,↓, (4.1)

where nr,σ ≡ c†r,σcr,σ, r is the set of sites on an arbitrary graph, (in D dimen-

sions) and 〈r, r′〉 denotes nearest neighboring sites. On a D-dimensions hypercu-

bic lattice with periodic boundary conditions and even lengths in all directions,

the Hubbard model admits translation invariance, charge and spin SU(2) sym-

metries, and lattice mirror symmetries. In that case, the Hubbard Hamiltonian

can be written as

HHub =∑k

∑σ∈↑,↓

Ekc†k,σck,σ + U∑r

nr,↑nr,↓, Ek ≡ −µ− 2tD∑i=1

cos ki, (4.2)

144

where ki is the momentum in the i-th direction. For these Hamiltonians, Refs. [185,

186] showed that there exists an operator η† defined as

η† ≡∑k

c†k,↑c†π−k,↓ =

∑r

eiπ·rc†r,↑c†r,↓, (4.3)

where π ≡ (π, π, · · · , π), that satisfies the relation

[HHub, η†] = (U − 2µ)η†. (4.4)

In fact, for a system with L sites in each dimension, the η† and η operators, along

with ηz ≡ 12[η†, η], constitute a full (pseudospin) SU(2) symmetry of the Hubbard

model [186]. That is,

[ηz, η†] = η†, [ηz, η] = −η, [HHub, ηz] = 0, [HHub,η

2] = 0, (4.5)

where η2 is the total pseudospin operator η2 ≡ 12(η†η + ηη†) + (ηz)

2.

Eq. (4.4) is said to be an example of a Spectrum Generating Algebra (SGA) [200,

201], when an operator satisfying∗

[H, η†] = Eη†, (4.6)

generates a series of equally spaced energy eigenstates. Indeed, if |ψ0〉 is an eigen-

state of H with energy E0, η† |ψ0〉 is also an eigenstate with energy E0 + E (see

App. D.3). Iterating this idea until (η†)N+1 |ψ0〉 vanishes (which it does, as η†

∗We note that the terms “Spectrum Generating Algebra” and “Dynamical Symmetry” havebeen used to denote a variety of related (but subtly distinct) concepts in the literature [202–205].In this work we will only use SGA to refer to Eq. (4.6).

145

increases the number of fermions by 2), we obtain an equally-spaced tower of

states given by |ψ0〉 , η† |ψ0〉 , · · · , (η†)N |ψ0〉 with corresponding energies given

by E0, E0 + E , E0 + 2E , · · · , E0 + NE. In the case of the Hubbard model,

E = (U − 2µ) (see Eq. (4.4)).

Although Eq. (4.6) leads to the existence of a tower of states starting from

|ψ0〉, it does not imply that the expressions of any of the eigenstates can be

obtained analytically. However, for the Hubbard Hamiltonian of Eq. (4.2) in any

dimensions, several U -independent eigenstates can be obtained analytically [186].

Note that the vacuum state |Ω〉, and the spin-polarized eigenstates of the hopping

operator are also ferromagnetic eigenstates of the Hubbard Hamiltonian. As a

consequence of the spin SU(2) symmetry, multiplets eigenstates can be obtained

by applying spin raising and lowering operators on these eigenstates. Further,

several more eigenstates are obtained applying the η† operator repeatedly on those

eigenstates, although not all of the resulting eigenstates are independent [186].

4.3 η-pairing on arbitrary graphs

To unravel the most general necessary conditions for having a spectrum generating

algebra, we break several symmetries of the Hubbard Hamiltonian HHub. We

consider much more general Hubbard Hamiltonians with disorder and spin-orbit

coupling on arbitrary graphs. This forces us to obtain a real-space understanding

of η-pairing, unlike the momentum space derivations used in most of the literature.

146

We consider the generalized Hubbard Hamiltonian

Hgen = −∑σ,σ′

∑⟨r,r′⟩

(tσ,σ

′

r,r′c†r,σcr′,σ′ + tσ

′,σr′,rc

†r′,σ′cr,σ

)︸ ︷︷ ︸

≡Tσ,σ′r,r′

−∑r,σ

µr,σnr,σ +∑r

Urnr,↑nr,↓,

(4.7)

where σ, σ′ denotes the spin, 〈r, r′〉 denote nearest neighboring sites r and r′ on

the graph, tσ,σ′

r,r′ are spin and position dependent hopping strengths satisfying

Hermiticity (tσ,σ′

r,r′ = tσ′,σ⋆

r′,r ), and µr,σ are spin dependent real chemical poten-

tials. Note that since we are considering the Hamiltonian Eq. (4.7) on arbitrary

graphs, we can without loss of generality consider the hopping terms tσ,σ′

r,r′ to be

non-vanishing on nearest neighboring sites on the graph. Note that the Hamilto-

nian of Eq. (4.7) breaks all the usual symmetries of the original Hubbard model

of Eq. (4.1) except the charge U(1) symmetry. Despite breaking these symme-

tries, we find that Hgen admits an SGA (and hence preserves the pseudospin “η”

symmetry) provided the hopping strengths tσ,σ′

r,r′, and µr,σ are appropriately

chosen. We define the η† operator to be

η† ≡∑r

qrη†r ≡

∑r

qrc†r,↑c

†r,↓, (4.8)

and derive conditions on qr and tσ,σ′

r,r′ such that the Hamiltonian of Eq. (4.7)

admits an SGA.

We first explicitly compute the following commutators of η† with the on-site

147

terms of the Hamiltonian

[∑r,σ

µr,σnr,σ, η†] =

∑r

qrµr,σ[nr,σ, η†r] =

∑r

qr

(∑σ

µr,σ

)η†r

[∑r,σ

Urnr,↑nr,↓, η†] =

∑r

qrUr[nr,↑nr,↓, η†r] =

∑r

qrUrη†r

(4.9)

where we have used Eq. (D.4). Thus, by choosing on-site chemical potentials and

interactions that satisfy Ur − µr,↑ − µr,↓ = E , we obtain

[∑r,σ

µr,σnr,σ +∑r,σ

Urnr,↑nr,↓, η†] = Eη†. (4.10)

Note that this allows for the addition of disordered on-site magnetic fields, see

Ref. [199] for an example of η-pairing in such a setting. We only require that E

does not depend on r.

We now move on to the hopping term in Eq. (4.7). In App. D.2, we show the

following (see Eq. (D.8))

[∑σ,σ′

T σ,σ′

r,r′ , qrη†r + qr′η†r′ ] = 0 (4.11)

provided tσ,σ′

r,r′ and qr satisfy (see Eq. (D.11))

qrtσ′,σr′,rsσ+qr′tσ,σ

′

r,r′sσ′ = 0 ∀σ, σ′, ∀〈r, r′〉, sσ ≡

+1 if σ =↑

−1 if σ =↓, σ ≡

↓ if σ =↑

↑ if σ =↓.

(4.12)

148

Using Eqs. (4.10) and (4.11), we obtain

[Hgen, η†] = Eη†, (4.13)

illustrating the generality of η-pairing.

4.4 Examples of η-pairing

We now discuss a few examples of η-pairing with and without spin-orbit coupling.

4.4.1 Without Spin-Orbit Coupling

We first consider the case without spin-orbit coupling. That is, we set

t↑,↓r,r′ = t↓,↑r,r′ = 0, ∀〈r, r′〉. (4.14)

Eq. (4.12) leads toqrqr′

= −t↓,↓r,r′

(t↑,↑r,r′)∗= −

t↑,↑r,r′

(t↓,↓r,r′)∗. (4.15)

Using Eq. (4.15), we obtain

|t↓↓r,r′ | = |t↑,↑r,r′ | =⇒|qr||qr′|

=|t↓,↓r,r′ ||t↑,↑r,r′ |

= 1. (4.16)

Without loss of generality, as the norm of qr is site-independent, we can set |qr| = 1

and choose

qr = eiϕr , t↑,↑r,r′ = tr,r′eiθ↑,↑

r,r′ , t↓,↓r,r′ = tr,r′eiθ↓,↓

r,r′ (4.17)

149

where tr,r′ is a positive real number, and

θ↑,↑r,r′ + θ↓,↓r,r′ + π = ϕr − ϕr′ . (4.18)

We now illustrate some examples of an SGA in a disordered system. For a

one-dimensional chain of length L, 1 ≤ r ≤ L the choice of hoppings θ↑,↑r,r′ =

θ↓,↓r,r′ = 0 corresponds to the usual Hubbard model of Eq. (4.1). Thus, according

to Eq. (4.18), we see that we can choose qr = eiπ·r when L is even for periodic

boundary conditions or any L for open boundary conditions, recovering the stan-

dard η† operator of Eq. (4.3). In fact, for bipartite graphs with sublattices A and

B, η† operators can be found for Hubbard models on by choosing

qr =

+1 if r ∈ A

−1 if r ∈ B, (4.19)

which has also been derived in Ref. [188].

To obtain η-pairing states on non-bipartite lattices, we could choose qr = ±1,

but we would necessarily have some pairs of nearest neighboring sites r and r′

such that ϕr = ϕr′ . Eq. (4.17) for such r and r′ can be satisfied by the choice

θ↑,↑r,r′ = θ↓,↓r,r′ = π2. For example, on a triangular lattice, for every triangle with

vertices denoted by r1, r2, r3 we could for example choose ϕr1 = ϕr3 = 0 and

ϕr2 = π. In such a case, we need to choose for example tr1,r2 = tr2,r3 = +t

and tr3,r1 = it to satisfy Eq. (4.18), which corresponds to the addition of a π/2

flux [196].

150

4.4.2 With Spin-Orbit Coupling

We now explore the case when hopping terms with spin-orbit coupling are added

to the generalized Hubbard Hamiltonian of Eq. (4.7). In addition to Eq. (4.14),

from Eq. (4.12) we obtain

qrtσ,σr′,r − qr′tσ,σr,r′ = 0 σ ∈ ↑, ↓, ∀〈r, r′〉, (4.20)

enforcing that

qrqr′

=t↑,↓r,r′(t↓,↑r,r′

)∗ =t↓,↑r,r′(t↑,↓r,r′

)∗ =⇒ |t↑,↓r,r′| = |t↓,↑r,r′ |. (4.21)

Thus, in addition to Eq. (4.17) we can choose t↑,↓r,r′ = tr,r′eiθ↑,↓

r,r′ , t↓,↑r,r′ = tr,r′eiθ↓,↑

r,r′

where tr,r′ is a positive real number such that

θ↑,↓r,r′ + θ↓,↑r,r′ = ϕr − ϕr′ . (4.22)

Thus, nearest-neighbor hopping terms with spin-orbit coupling to the Hubbard

model (θ↑,↑ = θ↓,↓ = 0) on a bipartite graph (so that the hoppings are always

between sites on different sublattices), provided they satisfy

θ↑,↓r,r′ + θ↓,↑r,r′ = ±π =⇒ t↑,↓r,r′ = −t↓,↑r,r′ , (4.23)

where we have used Eqs. (4.18) and (4.22). We can indeed verify that in this limit,

we recover the conditions derived for η-pairing in translation-invariant spin-orbit

coupled Hubbard models in Ref. [197].

151

4.5 Quantum Many-Body Scars from the Hubbard Model

As we showed in the previous sections, the existence of an SGA in the generalized

Hubbard models gives rise to several towers of η-pairing states. We now ask

if perturbations can be added to those models that preserve some but not all

of the towers generated by η-pairing are preserved. We introduce the concept

of a Restricted Spectrum Generating Algebra (RSGA), a restriction of the SGA

discussed in Sec. 4.2, and illustrate perturbations that realize those conditions.

These perturbed Hamiltonians hence preserve some towers generated by η-pairing,

and we argue that the resulting towers of eigenstates become quantum many-body

scars in the perturbed Hamiltonians. We note that everything we derive here

will apply to both the original Hubbard model of Eq. (4.2) and the generalized

Hubbard models of Eq. (4.7), but we focus on the latter for the sake of generality.

4.5.1 RSGA of Order 1

A Hamiltonian H is said to exhibit a Restricted Spectrum Generating Algebra

of Order 1 (RSGA-1) if there exists a state |ψ0〉 and an operator η† such that

η† |ψ0〉 6= 0 that satisfy

(i) H |ψ0〉 = E0 |ψ0〉 , (ii) [H, η†] |ψ0〉 = Eη† |ψ0〉 , (iii) [[H, η†], η†] = 0. (4.24)

As we show in Lemma D.3.2 in App. D.3, the conditions of Eq. (4.24) lead to the

existence of a equally-spaced tower of states (η†)n |ψ0〉 starting from |ψ0〉. We

illustrate this concept by choosing |ψ0〉 = |Ω〉, the empty vacuum state, and as a

152

perturbation of the Hamiltonian Hgen, the electrostatic interaction of the form

I2 ≡∑σ,σ′

∑⟨⟨r,r′⟩⟩

V σ,σ′

r,r′ nr,σnr′,σ′ , (4.25)

where 〈〈r, r′〉〉 runs over some or all pairs of sites on the graph. Note that this

sum can be restricted to only nearest-neighbor sites for a more physical choice of

interaction. Since |Ω〉 is an eigenstate of the Hubbard Hamiltonian Hgen and I2,

we obtain (Hgen + I2) |Ω〉 = 0, satisfying condition (i) of RSGA-1 with E0 = 0.

Further, using the commutation relation in Eq. (D.4)

[nr,σ, η†r] = η†r, (4.26)

we deduce for r 6= r′ that

[nr,σnr′,σ′ , qrη†r + qr′η†r′ ] = qrη

†rnr′,σ′ + qr′nr,ση

†r′ . (4.27)

Using Eq. (4.27), we note that

[nr,σnr′,σ′ , η†] |Ω〉 = [nr,σnr′,σ′ , qrη†r + qr′η†r′ ] |Ω〉 = 0. (4.28)

due to the r and r′ occupations of the vacuum state. As a consequence, we

obtain [I2, η†] |Ω〉 = 0 and [Hgen + I2, η

†] = Eη†, satisfying condition (ii) of RSGA-

1. Further, we note that using Eqs. (4.26) and (4.27), we obtain

[[nr,σnr′,σ′ , qrη†r + qr′η†r′ ], qrη

†r + qr′η†r′ ] = 2qrqr′η†rη

†r′ . (4.29)

153

Thus, we obtain

[[I2, η†], η†] = 2

∑⟨⟨r,r′⟩⟩

(∑σ,σ′

V σ,σ′

r,r′ )qrqr′η†rη†r′ . (4.30)

Setting ∑σ,σ′

V σ,σ′

r,r′ = 0, (4.31)

and using Eqs. (4.30) and (4.13), we obtain [[Hgen + I2, η†], η†] = 0, satisfying

condition (iii) of RSGA-1. Thus, as a consequence of Lemma D.3.2, the Hamil-

tonian (Hgen + I2) exhibits the tower (η†)n |Ω〉 as eigenstates, although other

η-pairing towers (starting from other states than the vacuum state) of the Hub-

bard Hamiltonian might not be preserved. A simple physical interaction that

satisfies Eq. (4.31) is the nearest neighbor Sz − Sz interaction.

4.5.2 RSGA of Order M

We now study perturbations to the Hubbard model that do not satisfy the con-

ditions of RSGA-1 but still preserve a tower of states. The concept of RSGA-1

can be generalized straightforwardly as follows. We define a set of states |ψn〉

as |ψn〉 ≡ (η†)n |ψ0〉 and a set of operators Hn as H0 ≡ H, and Hn+1 ≡ [Hn, η†]

A Hamiltonian H is said to exhibit a Restricted Spectrum Generating Algebra of

Order M (RSGA-M) if there exists a state |ψ0〉 and an operator η† such that

154

|ψn〉 6= 0 for n ≤M that satisfy

(i) H |ψ0〉 = E0 |ψ0〉 (ii) H1 |ψ0〉 = Eη† |ψ0〉 (iii) Hn |ψ0〉 = 0 ∀ n, 2 ≤ n ≤M

(iv) Hn

6= 0 if n ≤M

= 0 if n =M + 1, (4.32)

where condition (iii) of RSGA-1 of Eq. (4.24) has been modified. As we show

in Lemma D.3.3 in App. D.3, the conditions of Eq. (4.32) is equivalent to the

existence of a equally-spaced tower of states (η†)n |ψ0〉 starting from |ψ0〉. Note

that conditions (i)-(iii) of RSGA-M lead to the existence of exact eigenstates

|ψ0〉 , η† |ψ0〉 , · · · , (η†)M |ψ0〉 with energies E0, E0+E , · · · , E0+ME. If we do

not add condition (iv), then these are all the guaranteed eigenstates, for a given

M . Condition (iv) ensures that (η†)n |ψ0〉 is also an eigenstate of H for any n as

long as it does not vanish.

We now explicitly construct a perturbation to the generalized Hubbard model

Hgen that admits an RSGA of order M . Consider the (M + 1)-body density

interaction term

IM+1 ≡∑rj

Vσjrj

M+1∏j=1

nrj ,σj , (4.33)

where rj represent a (chosen) set of (M + 1) distinct sites and σj a set of

(M + 1) spins. On the vacuum state |Ω〉, we obtain (Hgen + IM+1) |Ω〉 = 0,

satisfying condition (i) of RSGA-M with E0 = 0. Using Eq. (4.26), we obtain

[M+1∏j=1

nrj ,σj , η†] =

M+1∑k=1

qrkη†rk

M+1∏j=1,j =k

nrj ,σj , (4.34)

and thus [IM+1, η†] |Ω〉 = 0. Using Eq. (4.13) we further obtain [Hgen+IM+1, η

†] |Ω〉 =

155

Eη† |Ω〉, satisfying condition (ii) of RSGA-M . Similarly, we can compute sub-

sequent commutators with η†. Since according to Eq. (4.26) each commutator

replaces an nr,σ by η†r, applying less than (M + 1) commutators, we obtain

[[M+1∏j=1

nrj ,σj , η†], η†] · · · ] · · · ]]︸ ︷︷ ︸

n times

6= 0 ∀ n, 2 ≤ n ≤M. (4.35)

Furthermore, since the commutator applied less than (M + 1) times necessarily

consists of at least one number operators nr,σ in each term, it vanishes on the

vacuum state |Ω〉, i.e.

[[M+1∏j=1

nrj ,σj , η†], η†] · · · ] · · · ]]︸ ︷︷ ︸

n times

|Ω〉 = 0 ∀ n, 2 ≤ n ≤M. (4.36)

The interaction IM+1 of Eq. (4.33) along with Eq. (4.13) thus satisfies conditions

(iii) of Eq. (4.32) with |ψ0〉 = |Ω〉. Applying the commutator (M + 1) times, we

obtain

[[M+1∏j=1

nrj ,σj , η†], η†] · · · ] · · · ]]︸ ︷︷ ︸

M+1 times

=M+1∏j=1

η†rj . (4.37)

Using Eqs. (4.33) and (4.37), we obtain

[[IM+1, η†], η†] · · · ] · · · ]]︸ ︷︷ ︸

M+1 times

=

∑rj,σj

Vσjrj

M∏j=1

η†rj . (4.38)

Condition (iv) of Eq. (4.32) can be satisfied if∑

rj,σjV

σjrj = 0 and the Hamilto-

nian (Hgen+ IM+1) exhibits an RSGA of order M . Note that while these perturba-

156

tions preserve the same tower of states (η†)n |Ω〉 as the perturbations illustrated

in Sec. 4.5.1, the algebra is different; this allows us to obtain many different terms

that can be added to the Hamiltonian in order to maintain this tower of states.

4.6 Connections to Quantum Scars

We now prove that the towers (η†)n |Ω〉 in the Hamiltonians (HHub + IM+1)

discussed as examples in Secs. 4.5.1 and 4.5.2 are the quantum many-body scars

for appropriate values of the Hamiltonian parameters, when it is non-integrable

and expected to satisfy ETH. Physically, the states of the tower are composed of

doubly-occupied quasiparticles “doublons” dispersing on top of the vacuum state

|Ω〉. The energy of a doublon is (U − 2µ) under the Hubbard Hamiltonian, hence

the state (η†)n |Ω〉 has an energy n(U − 2µ), spin Sz = 0, and charge Q = 2n.

The highest state of the tower consists of all the sites being filled with doublons.

As expected for quasiparticles on top of a low entanglement state [56, 57], and

as rigorously computed in Ref. [152], the entanglement entropy S of the state

(η†)n |Ω〉 scales as the logarithm of the subsystem volume (S ∼ log V ), in contrast

to the volume-law (S ∼ V ) predicted by ETH [18]. By estimating the energies

of the states in the sector with spin Sz = 0 and charge Q = 2n, we now show

that some states of the tower (η†)n |Ω〉 can be in the bulk of the spectrum of

their quantum number sectors. Note that, unlike in Ref. [152], we have now lost

the η-pairing symmetry, and hence the theorem, proven in Ref. [152] - that the η-

pairing states are the only ones in their quantum number sectors, does not apply.

We do this in one and higher dimensions separately for pedagogical purposes.

157

4.6.1 One Dimension

For example, consider the Hubbard model HHub of Eq. (4.1) in one dimension with

an even system size L. Non-interacting ferromagnetic eigenstates with charge

Q = 2n can be constructed by occupying the single-particle spectrum of the

quadratic part of the Hamiltonian HHub with 2n ↑ spins. These eigenstates have

spin quantum numbers Sz = n, and the lowest and highest energies E− and E+

of such states are given by

E± = −2nµ ± 2tn−1∑j=−n

cos

(2πj

L

)= −2nµ ± 2t cot

(πL

)sin

(2nπ

L

). (4.39)

As a consequence of the spin-SU(2) symmetry of HHub, eigenstates with the same

energies but with spin Sz = 0 can be constructed by applying the spin lowering

operator on these non-interacting ferromagnetic eigenstates. By adding a pertur-

bation IM+1 that breaks spin-flip symmetry and translation invariance to HHub,

we break the integrability of the one-dimensional Hubbard model † and all the

symmetries of HHub except spin Sz and charge U(1). For a small perturbation

strength, we expect the lowest and highest energy eigenstates of the Sz = 0

and Q = 2n sector to still be upper and lower bounded by (approximately) E−

and E+ respectively. Thus, we expect the state (η†)n |Ω〉 to be certainly in the

bulk of the spectrum of its own quantum number sector Sz = 0, Q = 2n if†We have numerically checked that the Hamiltonians (HHub + I2) with nearest-neighbor

electrostatic interactions exhibit level repulsion and GOE level statistics for generic values ofcouplings V σ,σ′

r,r′ , even if they satisfy Eq. (4.31).

158

E− < n(U − 2µ) < E+, or,

−2t

ncot

(2π

L

)sin

(2nπ

L

)< U <

2t

ncot

(2π

L

)sin

(2nπ

L

), (4.40)

which can always be satisfied by an appropriate choice of U and t. For a finite

density of doublons in the thermodynamic limit (n/L = ρ and n, L→∞), using

Eq. (4.40) we obtain

−sin (2πρ)

πρ<U

t<

sin (2πρ)

πρ. (4.41)

We could also add small spin-orbit coupling and disorder to HHub to obtain Hgen.

This breaks the spin U(1) symmetry, which combines the quantum number sectors

of various Sz’s with the same Q. The estimate of Eq. (4.41) is thus a condition

under which some states of the tower (η†)n |Ω〉 are quantum many-body scars

of the Hamiltonian (Hgen + IM+1). While we have broken translation invariance

here, it is easy to show that these scars are in the bulk of the spectrum as long as

Eq. (4.41) is satisfied, even when translation, inversion, and spin-flip symmetries

are not broken.

4.7 D dimensions

In this appendix, we obtain the conditions for which the states of the tower

(η†)n |Ω〉 are in the bulk of the spectrum of the Hamiltonian (HHub + IM+1)

in D-dimensions. Consider a system of L×L×· · ·×L sites in D dimensions, and

the state (η†)n |Ω〉 consisting of n doublons. To obtain the conditions for (η†)n |Ω〉

to lie in the bulk of the spectrum for a small perturbation IM+1, it is sufficient

159

to obtain the energies E− and E+ of the lowest and highest ferromagnetic non-

interacting eigenstates with charge Q = 2n, as discussed in Sec. 4.6. To do so,

we directly work in the continuum limit in momentum space and with a finite

density of doublons, i.e. ρ ≡ nLD . The single-particle density of states f(k) reads

f(k) =

(L

2π

)D,

∫dDk f(k) = LD. (4.42)

Assuming a spherical Fermi surface, the Fermi momentum kF by filling the lowest

2n single-particle levels satisfies the relation

∫|k|<kF

dDk f(k) =

(L

2π

)Dπ

D2 kDF

Γ(D2+ 1) = 2n, (4.43)

where we have used the expression for the volume of D-dimensional sphere. We

thus obtain

kF =2√π

L

(2n Γ

(D

2+ 1

)) 1D

= 2√π

(2ρ Γ

(D

2+ 1

)) 1D

. (4.44)

Note that the spherical approximation of the Fermi surface in Eq. (4.43) breaks

down for sufficiently large n when the Fermi surface is close to the edges of the

Brillouin zone, i.e. when the Fermi momentum kF obtained using Eq. (4.44) is

comparable to π. Thus, the calculations in this section are strictly valid only when

kF π, or when the doublon density ρ satisfies

ρ πD2

2D+1Γ(D2+ 1) . (4.45)

160

However, we expect that similar arguments work for the larger densities as well.

Using the dispersion relation of the quadratic part of the D-dimensional Hubbard

models, the energy of the states obtained by filling all single-particle momentum

levels with |k| < kF is given by

E− = −2nµ− 2tD

(L

2π

)D ∫|k|<kF

dDk cos (ki)︸ ︷︷ ︸≡ID(kF )

, (4.46)

where ki is the component of k along any axis. Evaluating the integral ID(kF ) in

D-dimensional spherical coordinates, we obtain

ID(kF ) = (2πkF )D2 JD

2(kF ), (4.47)

where Jα(x) is the α-th order Bessel function of the first kind. Note that Jα(x)

for α ∈ Z + 12

can be expressed in terms of trigonometric functions. Thus, for

D = 1 for example, we obtain I1(kF ) = 2 sin(kF ). For the highest energy state,

similar to Eq. (4.46), we obtain

E+ = −2nµ+ 2tD

(L

2π

)DID(kF ). (4.48)

The state (η†)n |Ω〉 with n doublons is thus guaranteed to be in the bulk of the

spectrum if E− < n(U − 2µ) < E+, or,

− 2D

(2π)DID(kF )ρ

<U

t<

2D

(2π)DID(kF )ρ

, (4.49)

161

where ID(kF ) and kF are defined in Eqs. (4.47) and (4.44) respectively. Note that

we recover Eq. (4.41) by setting D = 1 in Eq. (4.49). For D = 2, the bound reads

−√

8

πρJ1(√

8πρ) <U

t<

√8

πρJ1(√

8πρ). (4.50)

4.8 RSGA and Quantum Scarred Models

Exact towers of states as discussed in Sec. 4.5 are also found in several models

of exact quantum many-body scars [55–59, 61, 63]. In this section, we briefly

comment on the connections between the RSGA formalism introduced here and

quantum scarred models in the literature, and in particular the unified formalism

introduced in Ref. [61]. The theorem of Eq. (1) in Ref. [61] states that given an

eigenstate |ψ0〉 of Hamiltonian H with energy E0, and a subspace W such that

|ψ0〉 ∈ W , a tower of equally spaced states (η†)n |ψ0〉 with energies E0 + nE

is guaranteed if for any |ψ〉 ∈ W

(i) [H, η†] |ψ〉 = Eη† |ψ〉 , (ii) η† |ψ〉 ∈ W. (4.51)

Since the RSGAs guarantee the existence of a tower of states (η†)n |ψ0〉, they sat-

isfy the conditions of Eq. (4.51) by choosing the subspaceW = span|ψ0〉 , η† |ψ0〉 , · · · , (η†)N |ψ0〉,

and are captured by the formalism of Ref. [61]. However, since we can obtain RS-

GAs of any order, they provide a finer classification of quantum scarred models.

We illustrate this connection by focusing on two examples: (i) the spin-1 XY model

family studied in Ref. [57], which we find admit RSGAs of order M = 1, and (ii)

the families of spin-1 scarred Hamiltonians (including the AKLT Hamiltonian)

162

studied in Refs. [55, 61, 63], which we find admit RSGAs of order M = 2.

4.8.1 Spin-1 XY Model

The spin-1 XY Hamiltonian family on a D-dimensional hypercubic lattice with

size L in each dimension is given by

H(x) = J∑⟨r,r′⟩

(SxrSxr′ + SyrS

yr′)︸ ︷︷ ︸

HXY

+h∑r

Szr︸ ︷︷ ︸Hz

+D∑r

(Szr)2

︸ ︷︷ ︸Hz2

. (4.52)

Throughout this section we label the spin-1 degrees of freedom by +, 0,−. As

discussed in Ref. [57], the spin-1 XY Hamiltonian has a tower of states starting

from a spin-polarized root eigenstate |Ω〉 ≡ |− − · · · − −〉:

H(x)(P(x))n |Ω〉 = (h(2n− LD) +DLD)(P(x))n |Ω〉 , (4.53)

for 0 ≤ n ≤ LD and P(x) ≡∑r

eiπ·r(S+r )

2. Ref. [61] showed that

[Hz,P(x)] = 2hP(x), [Hz2 ,P(x)] = 0, [HXY ,P(x)] = 4J∑⟨r,r′⟩

eiπ·rhr,r′ , (4.54)

where hr,r′ = |0 +〉 〈− 0| − |+ 0〉 〈0 −|. We can thus decompose H(x) as

H(x) = Hz2 +Hz︸ ︷︷ ︸H

(x)SGA

+HXY︸︷︷︸V (x)

, (4.55)

where H(x)SGA admits an exact SGA, i.e., [H(x)

SGA,P(x)] = E (x)P(x) with E (x) = 2h.

We thus obtain [H(x),P(x)] = 2hP(x)+4J∑

⟨r,r′⟩eiπ·rhr,r′ . Noting that hr,r′ |Ω〉 = 0,

163

we obtain [H(x),P(x)] |Ω〉 = 2hP(x) |Ω〉. Further, we also obtain

[[H(x),P(x)],P(x)] = 4J∑⟨r,r′⟩

[hr,r′ , (S+r )

2 − (S+r′)

2] = 0. (4.56)

Thus, the family of Hamiltonians of Eq. (4.53) admit an RSGA of order M = 1

(see Lemma D.3.2 in App. D.3) with |ψ0〉 = |Ω〉, E0 = (D − h)LD, E = 2h,

η† = P(x).

4.8.2 Spin-1 AKLT Model

In this section, we show that the AKLT family of quantum scarred Hamiltonians

studied in Chapter 3 admit RSGAs of order M = 2. The one-dimensional family

of quantum scarred spin-1 Hamiltonians (including the spin-1 AKLT chain) on a

system size of L is given by

H(a) =L∑j=1

hj,j+1, hj,j+1 = E (|J2,1〉 〈J2,1|+ |J2,2〉 〈J2,2|)+0∑

m,n=−2

z(m,n)j (|J2,m〉 〈J2,n|).

(4.57)

where (z(m,n)j ) = (z

(m,n)j )∗. As discussed in Chapter 3, for an even system size L

and periodic boundary conditions, the Hamiltonian of Eq. (4.57) contains a tower

of quantum scars from a root eigenstate |G〉,

H(a)(P(a))n |G〉 = 2nE(P(a))n |G〉 , 0 ≤ n ≤ L

2, (4.58)

where |G〉 is the spin-1 AKLT ground state, and P(a) =L∑j=1

(−1)j(S+j )

2, which

forms the analogue of the η† operator in the Hubbard models discussed in the

164

main text. We first compute the commutator

[H(a),P(a)] =L∑j=1

(−1)j[hj,j+1, (S+j )

2 − (S+j+1)] ≡ 2EP(a) +

L∑j=1

(−1)jh(1)j,j+1,

h(1)j,j+1 = −2

0∑n=−2

((z

(−1,n)j − E) |J1,1〉 〈J2,n|+

√2(z

(−2,n)j − E) |J1,0〉 〈J2,n|

),

(4.59)

where we have used Eq. (4.57), and [61]

(S+j )

2−(S+j+1)

2 = −2(|J2,1〉 〈J1,−1|+

√2 |J2,2〉 〈J1,0|+ |J1,1〉 〈J2,−1|+

√2 |J1,0〉 〈J2,−2|

).

(4.60)

Using Eqs. (4.57) and (4.59), it is apparent that H(a) can be decomposed as

H(a) = H(a)SGA + V (a), where

H(a)SGA ≡

L∑j=1

E (|J2,1〉 〈J2,1|+ |J2,2〉 〈J2,2| − |J2,−1〉 〈J2,−1| − |J2,−2〉 〈J2,−2|),(4.61)

and it admits an exact SGA, i.e. [H(a)SGA,P(a)] = 2EP(a). Further, we note that

h(1)j,j+1 |G〉 = 0, since the AKLT ground state does not have a total spin 2 compo-

nent over neighboring sites. Using Eq. (4.59), we thus obtain [H(a),P(a)] |G〉 =

2EP(a) |G〉. We further compute the next commutator

[[H(a),P(a)],P(a)] =L∑j=1

[h(1)j,j+1, (S

+j )

2 − (S+j+1)

2] ≡L∑j=1

h(2)j,j+1

h(2)j,j+1 = −4

√2

0∑n=−2

(z(−2,n)j − E) |J2,2〉 〈J2,n|. (4.62)

165

We thus obtain h(2)j,j+1 |G〉 = 0 and [[H(a),P(a)],P(a)] |G〉 = 0 and further,

[[[H(a),P(a)],P(a)],P(a)] =L∑j=1

(−1)j[h(2)j,j+1, (S+j )

2 − (S+j+1)

2] = 0. (4.63)

Thus, the family of Hamiltonians of Eq. (4.57) admit an RSGA of order M = 2

(see Lemma D.3.3) with |ψ0〉 = |G〉, E0 = 0, E = 2E , and η† = P(a). Similarly, we

can verify that the same algebraic structure holds for the single-site quasiparticle

family of scarred Hamiltonians studied in Ref. [63], the one-dimensional spin-S

AKLT Hamiltonians discussed in Chapter 1 and the associated family of scarred

Hamiltonians discussed in Ref. [61].

4.9 Conclusions

In this chapter, we have shown how quantum many-body scars based on η-pairing

states can appear in generalized and perturbed fermionic Hubbard models. We

have explored the η-pairing states in the Hubbard model and generalized in two

directions. First, casting η-pairing as a real-space phenomenon, we find a highly

general Hubbard Hamiltonian potentially with disorder and spin-orbit coupling

that exhibits a Spectrum Generating Algebra (SGA). Second, we introduce the

concept of Restricted SGA (RSGA) and add use it to find various perturbations

to the (generalized) Hubbard models that preserve the η-pairing tower starting

from the vacuum state. The states of this tower have a sub-thermal entanglement

entropy, and we analytically obtain conditions for the states of this tower to lie in

the bulk of the spectrum of their quantum number sector, showing that they are

examples of quantum many-body scars. We further connected RSGAs to some

166

models of exactly solvable quantum scars in the literature, particularly the first

two examples of towers of quantum scars in the AKLT model [55] and the spin-1

XY model [57]. The scars there can thus be explained by the existence of RSGAs

obtained by perturbing Hamiltonians with exact SGAs.

There are many natural extensions to this work. It is important to understand

the connection of RSGAs with models of quantum scars that exhibit multi-site

quasiparticles [58,59,63], including Hubbard models with generalized η-pairing [152],

where the discussions in this work do not seem to generalize easily. Further, the

RSGAs described here closely resemble algebraic structures introduced in ear-

lier works both in the context of ground states [206, 207] as well as quantum

scars [61, 62], and it is highly desirable to better understand the connections be-

tween them, and also connections to the embedding construction in Ref. [181]. Ap-

propriate generalizations of RSGAs might also provide a way to construct closed

solvable subspaces that are not necessarily equally spaced towers of states, akin to

the closed Krylov subspaces found in several constrained systems [86–88,208]. On

a different note, given that the SGAs and RSGAs survive in the presence of disor-

der, it would be interesting to understand the existence and implications of these

towers of states in the many-body localized regime in Hubbard models [209–213].

167

5Hilbert Space Fragmentation

5.1 Introduction

In this chapter, we investigate the quantum dynamics of a translation invariant,

non-integrable 1D fermionic chain with conserved center-of-mass (CoM). Rather

than imposing constraints by hand, we show that the CoM conserving model we

study has a natural origin in two distinct physical settings: in the thin-torus

limit of the fractional quantum Hall effect and in the strong electric field limit

168

of the interacting Wannier-Stark problem, a regime accessible to current cold-

atom experiments. Focusing on systems close to half-filling, we define composite

degrees of freedom in terms of which CoM conservation maps onto dipole moment

conservation, revealing the underlying fractonic nature of the model.

Once we resolve the Hamiltonian into its disparate symmetry sectors, we find

that the Hilbert space further shatters into exponentially many dynamically dis-

connected sectors or Krylov subspaces, which have previously been studied under

various settings [78, 85–87, 208]. This shattering is a consequence of charge and

center-of-mass conservation and, as discussed in Refs. [86, 87], the presence of

exponentially many small (finite size in the thermodynamic limit) closed Krylov

subspaces can lead to effectively localized dynamics. Here, we instead focus on

a new phenomenon within exponentially large Krylov subspaces, which are of in-

finite size in the thermodynamic limit, and unveil a rich structure within these

sectors, leading to new notions of Krylov-restricted integrability and thermaliza-

tion.

Specifically, we find that several such large Krylov subspaces are integrable,

thereby establishing the phenomenon of emergent integrability and further break-

ing of ergodicity within closed Krylov sectors. Meanwhile, other large sectors

remain non-integrable. To bring this distinction into focus, we propose that a

modified version of ETH applies to Krylov fractured systems, wherein conven-

tional diagnostics of non-integrability, such as level statistics, are defined with

respect to a symmetry sector and a Krylov subspace. Using our modified def-

inition, we conclude that the problem ‘thermalizes’ within each non-integrable

Krylov subspace, in that the long-time behaviour of a state belonging to a par-

ticular Krylov subspace coincides with the Gibbs ensemble restricted to that sub-

169

space. Remarkably, we find that this restricted thermalization within some of the

Krylov subspaces leads to the ‘infinite temperature’ state within the Krylov sec-

tors showing atypical behaviour, in that the late-time charge density deviates from

that expected from unconstrained translation invariant systems. Violations of this

modified or ‘Krylov-restricted ETH’ require either integrability, conventional ‘dis-

order induced’ many-body localization, or existence of further symmetries within

the Krylov subspace.

This chapter is organized as follows: we introduce the pair-hopping model, in

Sec. 5.2 and show that it conserves center-of-mass. We then briefly discuss its

origins in the thin torus limit of the fractional quantum Hall effect (FQHE) and

in the limit of strong electric field in the interacting Wannier-Stark problem. In

Sec. 5.3, we introduce a convenient formalism to study this model at half-filling,

and show that it exhibits fractonic phenomenology. In Sec. 5.4, we discuss the no-

tion of Krylov fracture i.e., the phenomenon where systems exhibit several closed

subspaces that are dynamically disconnected with respect to product states. We

show examples of integrable and non-integrable dynamically disconnected Krylov

subspaces in Secs. 5.5 and 5.6 respectively. The integrable subspaces we study

exactly map onto XX models of various sizes, and the non-integrable subspaces

show features that are typically not expected in non-integrable models, which we

discuss in Sec. 5.7.

5.2 Model and its symmetries

The “pair-hopping model” we study is a one-dimensional chain of interacting

spinless fermions with translation and inversion symmetry, with the Hamilto-

170

nian [78, 214]

H =

Lb∑j=1

Hj =

Lb∑j=1

(c†jc

†j+3cj+2cj+1 + h.c.

), (5.1)

where Lb = L − 3 for open boundary conditions (OBC), Lb = L for periodic

boundary conditions (PBC), and the subscripts are defined modulo L for PBC.

Note that we have set the overall energy scale equal to one for convenience. Each

term Hj of Eq. (5.1) vanishes on all spin configurations on sites j to j + 3 except

for

Hj

j j+3

|0 1 1 0〉 =j j+3

|1 0 0 1〉, Hj

j j+3

|1 0 0 1〉 =j j+3

|0 1 1 0〉, (5.2)

where |a b c d〉 represents the occupation of sites j to j + 3. In the rest of this

chapter, we will use the following shorthand notation |1 0 0 1〉 ↔ |0 1 1 0〉 to

represent Eq. (5.2) i.e., the action of individual terms of the Hamiltonian Eq. (5.1).

This pair-hopping model preserves the center-of-mass position i.e., the center-of-

mass position operator [214]

C ≡

L∑j=1

jnj if OBC

exp

(2πiL

L∑j=1

jnj

)if PBC

, (5.3)

where the number operator nj ≡ c†jcj commutes with the Hamiltonian of Eq. (5.2).

Hamiltonians with such conservation laws, including the model given by Eq. (5.1),

were first discussed in Ref. [214] in the quest to build featureless Mott insulators.

As emphasized by Ref. [214], the spectra of center-of-mass preserving Hamilto-

nians have some unusual features. For example, at a filling ν = p/q (with p and

q coprime), the full spectrum is q-fold degenerate, which stems from the fact that

171

the center-of-mass position operator C, and the translation operator T do not

commute. More precisely, consider a 1D chain of length L with periodic bound-

ary conditions. As shown in Ref. [214], CT = e2πiνT C, where ν is the the filling

fraction ν = p/q. This results in a q-fold degeneracy of the spectrum with PBC.

The pair-hopping model Eq. (5.1), with even system size L = 2N and with

PBC, has an additional symmetry: sublattice particle number conservation. That

is, the operators

ne =N∑j=1

n2j, no =N−1∑j=1

n2j+1 , (5.4)

both commute with Eq. (5.1). This can be seen by writing the action of the terms

of the pair-hopping Hamiltonian as

e o e o

|1 0 0 1〉 ↔e o e o

|0 1 1 0〉,o e o e

|1 0 0 1〉 ↔o e o e

|0 1 1 0〉, (5.5)

where the superscripts o and e label the parity of the sites. The actions of

Eq. (5.5) conserve the particle number on the odd and even sites separately.

Sublattice number conservation of Eq. (5.4) trivially implies the conservation

of total particle number (ne + no). Note that the sublattice number conserva-

tion is a special property of the truncated Hamiltonian Eq. (5.1), and does not

hold in general for center-of-mass preserving Hamiltonians. For example, the ex-

tended pair-hopping Hamiltonian∑j

(c†jc

†j+3cj+2cj+1 + c†jc

†j+4cj+3cj+1 + h.c.

)pre-

serves the center-of-mass position but does not conserve sublattice particle num-

ber.

172

Experimental Relevance

An especially appealing feature of center-of-mass preserving terms, including the

pair-hopping term Eq. (5.1), is their natural appearance in multiple experimentally

relevant systems. The first setting in which such models appear is in the quantum

Hall effect, when translation invariant interactions are projected onto a single

Landau level [78, 215, 216]. We refer the reader to Appendix E for a derivation,

but summarize the general idea here: one works in the Landau gauge, such that

the single particle orbitals in a Landau level can be written as eigenstates of the

magnetic translation operators in the y direction, in which case the position in

the x direction is the momentum quantum number in the y direction. The matrix

elements of a translation invariant interaction between the single particle orbitals

are hence momentum conserving in the y direction, which translates to center-of-

mass conservation in the x direction of the effective one-dimensional model [215].

A general interaction operator projected to a Landau level of an Lx×Ly quantum

Hall system has the form

H =

NΦ∑j=1

∑k,m

Vkm

(c†jc

†j+k+mcj+kcj+m + h.c.

), (5.6)

whereNΦ = LxLy/ (2π) is the number of flux quanta and Vkm ∼ exp(−2π2 (k2 +m2) /L2

y

)with the magnetic length set to unity. Thus, in the “thin-torus” limit (Ly → 0),

one of the dominant terms is the pair-hopping Hamiltonian Eq. (5.1). We note

that such Hamiltonians also appear in the thin torus limit of the pseudopotential

Hamitonians for several Fractional Quantum Hall states [78, 217–220].

A second origin of such center-of-mass preserving models is in the well-known

173

Wannier-Stark problem [221]: spinless fermions hopping on a finite one-dimensional

lattice, subject to an electric field. While localization at the single-particle level

has been long established [222], an interacting version of the problem has recently

been studied and found to display behaviour associated with MBL systems at

strong fields [223, 224]; this phenomenon goes under the name Bloch (or Stark)

MBL. In Appendix E, we show that the dynamics of the Bloch MBL model in the

limit of an infinitely strong electric field is governed by an effective center-of-mass

preserving Hamiltonian, with the lowest order “hopping” term given precisely by

Eq. (5.1). Specifically, the resulting Hamiltonian is again of the form Eq. (5.6),

with NΦ replaced by the system size∗. This mapping hence allows us to present

a new perspective on the phenomenon of Bloch MBL (see Appendix E), in addi-

tion to providing a natural experimental setting, accessible to current cold-atom

experiments, for realizing the model studied here.

5.3 Hamiltonian at 1/2 filling

We now proceed to study the spectrum of the pair hopping Hamiltonian Eq. (5.1).

In this work, we will be focusing on systems at, or close to, half filling, and will

restrict ourselves to even system sizes L = 2N . For the study of this Hamiltonian

at other filling factors, see Refs. [78, 225].∗Note that for both the FQHE and the Bloch MBL case, the dominant center-of-mass conserv-

ing terms are nearest neighbor (nj nj+1) and next nearest neighbor electrostatic terms (nj nj+2),but the lowest order “hopping” is the pair-hopping Hamiltonian of Eq. (5.1).

174

5.3.1 Composite degrees of freedom

To study this model, and to elucidate its relation to the physics of fractons, we

define composite degrees of freedom formed by grouping neighboring sites of the

original model. Assuming an even number of sites, we group sites 2j−1, 2j of the

original lattice into a new site j so as to form a new chain with N = L/2 sites.

We define new degrees of freedom for these composite sites as follows:

|↑〉 ≡ |0 1〉 , |↓〉 ≡ |1 0〉 , |+〉 ≡ |1 1〉 , |−〉 ≡ |0 0〉 . (5.7)

The choice of grouping is unambiguously defined for OBC, and we stick to it for

most of this chapter. Writing the action of the Hamiltonian Eq. (5.2) in terms of

these composite degrees of freedom, we find

∣∣∣ 01 10⟩↔∣∣∣ 10 01

⟩⇐⇒ |↑↓〉 ↔ |↓↑〉 , (5.8)∣∣∣ 10 11 00

⟩↔∣∣∣ 11 00 10

⟩⇐⇒ |↓ +−〉 ↔ |+− ↓〉 , (5.9)∣∣∣ 00 11 01

⟩↔∣∣∣ 01 00 11

⟩⇐⇒ |−+ ↑〉 ↔ |↑ −+〉 , (5.10)∣∣∣ 10 11 01

⟩↔∣∣∣ 11 00 11

⟩⇐⇒ |↓ + ↑〉 ↔ |+−+〉 , (5.11)∣∣∣ 01 00 10

⟩↔∣∣∣ 00 11 00

⟩⇐⇒ |↑ − ↓〉 ↔ |−+−〉 , (5.12)

where · · · represents a grouping of some sites 2j − 1 and 2j, and |a〉 ↔ |b〉

represents the action of a single term of the Hamiltonian on |a〉 resulting in |b〉

and vice versa (see Eqs. (5.2)). For reasons that will become clear forthwith, we

set the nomenclature of the composite degrees of freedom as follows:

|+〉, |−〉: Fractons |+−〉, |−+〉: Dipoles |↑〉, |↓〉: Spins

175

Here, Eqs. (5.9)-(5.12) resemble the rules restricting the mobility of fractons, and

are similar to those discussed in Ref. [98] (see Ref. [226] for a review on fractons).

In particular, Eqs. (5.9) and (5.10) represent the free propagation of dipoles

when separated by spins, and Eqs. (5.11) and (5.12) encode the characteristic

movement of a fracton through the emission or absorption of a dipole, i.e. dipole

assisted hopping. However, in contrast to usual fracton phenomenology, here the

movement of fractons is also sensitive to the background spin configuration. For

example, the fracton in the configuration |· · · ↓ + ↑ · · · 〉 can move by emitting

a dipole (see Eq. (5.11)) while that in the configuration |· · · ↑ + ↓ · · · 〉 cannot.

In our convention, the fractons |+〉 and |−〉 have spin 0 and charges +1 and

−1 respectively, while the spins |↑〉 and |↓〉 have charge 0 and spins +1 and −1

respectively. Thus the unit cell charge and spin operators in terms of the original

fermionic degrees of freedom read

Qj ≡ n2j−1 + n2j − 1, Szj ≡ −n2j−1 + n2j, (5.13)

where j is the unit cell index, and 2j − 1, 2j are the site indices of the original

configuration. We represent the total number of +, −, ↑, and ↓ by N+, N−, N↑,

N↓ respectively. Thus, the total charge is N+−N− and the total spin is N↑−N↓.

5.3.2 Symmetries in terms of the composite degrees

We now study the symmetries of the Hamiltonian whose terms act on the com-

posite degrees of freedom through Eqs. (5.8)-(5.12). As discussed in Sec. 5.3, the

pair-hopping model Eq. (5.1) has several symmetries: sublattice charge conser-

vation, center-of-mass conservation, inversion, and translation (for PBC). Using

176

Eqs. (5.8)-(5.12), we now interpret these symmetries in terms of the composite

degrees of freedom defined in Eq. (5.7).

The model in terms of the composite degrees of freedom conserves the total spin

and the total charge, as is evident from Eqs. (5.8)-(5.12). In other words, N↑−N↓

and N+ −N− are separately conserved. Indeed, using the definitions of spin and

charge in Eq. (5.13), the total spin operator Sz and total charge operator Q can

be expressed in terms of the operators in the original Hilbert space as follows:

Q ≡N∑j=1

Qj = ne + no −N, Sz ≡N∑j=1

Szj = no − ne, (5.14)

where ne and no are the sublattice particle numbers defined in Eq. (5.4). Thus,

the conservation of total charge and total spin in the fracton model is a direct

consequence of the sublattice number conservation of the pair-hopping model.

Moreover, the fractonic behavior inherent in the rules specified by Eqs. (5.8)-

(5.12) suggests that the dipole moment of the composite degrees of freedom is a

conserved quantity [227]. This operator is defined similarly to the center-of-mass

operator Eq. (5.3) as:

D ≡

N∑j=1

jQj if OBC

exp

(i2πN

N∑j=1

jQj

)if PBC

. (5.15)

To explicitly show that D is in fact a conserved quantity of the composite fractonic

177

model, we observe that

N∑j=1

jQj =N∑j=1

j (n2j−1 + n2j − 1) =1

2

L∑j=1

jnj +no2− N(N + 1)

2. (5.16)

Then, using Eqs. (5.3), (5.4), and (5.16), in terms of the original operators in the

pair-hopping model, the operator D can be expressed as

D =

12C + 1

2no − N(N+1)

2if OBC

C12 ei

πLnoe−i

πN(N+1)L if PBC

. (5.17)

Since C and no are conserved operators of the pair-hopping Hamiltonian, as dis-

cussed in Sec. 5.2, it follows from Eq. (5.17) that D is conserved in the composite

model. To complete our discussion, we note that the composite model also pre-

serves inversion as well as translation symmetry (with PBC), neither of which

commute with D.

5.4 Krylov Fracture

We now study the dynamics of H, and show that it exhibits exponentially many

dynamically disconnected subspaces. More precisely, we construct Krylov sub-

spaces of the form

K (H, |ψ0〉) ≡ span|ψ0〉 , H |ψ0〉 , H2 |ψ0〉 , · · · (5.18)

that are by definition closed under the action of the Hamiltonian H. While |ψ0〉

in Eq. (5.18) can in principle be an arbitrary state, we are interested in the

178

dynamics of initial product states, which are more easily accessible to experiments.

Hence, we focus on Krylov subspaces generated by product states |ψ0〉, which we

dub root states of the Krylov subspace K (H, |ψ0〉). For a generic non-integrable

Hamiltonian H without any symmetries, one expects that K (H, |ψ0〉) for any

initial product state |ψ0〉 is the full Hilbert space of the system. For a non-

integrable Hamiltonian with some symmetry, and with |ψ0〉 an eigenstate of the

symmetry, one typically expects that K (H, |ψ0〉) spans all states with the same

symmetry quantum number as |ψ0〉.

Surprisingly, however, we show that the pair-hopping Hamiltonian (5.1) exhibits

Krylov fracture i.e., even after resolving the charge and center-of-mass symmetries,

we find generically that K (H, |ψ0〉) does not span all states with the same sym-

metry quantum numbers as |ψ0〉. Thus the full Hilbert space of the system H is

of the form

H =⊕s

H(s), H(s) =K(s)⊕i=1

K(H,∣∣∣ψ(s)

i

⟩), (5.19)

where s labels the distinct symmetry quantum numbers, such as charge and center-

of-mass, K(s) denotes the number of disjoint Krylov subspaces generated from

product states with the same symmetry quantum numbers, and∣∣∣ψ(s)

i

⟩are the root

states generating the Krylov subspaces. Note that the root states in Eq. (5.19)

are chosen such that they generate distinct disconnected Krylov subspaces, since

the same subspace can be generated by different root states. Stated symbolically,

K(H,∣∣∣ψ(s)

i

⟩)∩ K

(H,∣∣∣ψ(s′)

i′

⟩)= δs,s′δi,i′K

(H,∣∣∣ψ(s)

i

⟩). (5.20)

Fracture of the form Eq. (5.19), where the total number of Krylov subspaces

K(s) is exponentially large in the system size, was recently shown to always exist in

179

Hamiltonians and random-circuit-models with center-of-mass conservation [86,87]

(alternatively referred to as “dipole moment” conservation). While the presence of

these symmetries guarantees fracture, one can distinguish between “strong” and

“weak” fracture [86,87], depending respectively on whether or not the ratio of the

largest Krylov subspace to the Hilbert space within a given global symmetry sector

vanishes in the thermodynamic limit. Strong (resp. weak) fracture is associated

with the violation of weak (resp. strong) ETH with respect to the full Hilbert

space. The pair-hopping model Eq. (5.1) (which is equivalent to the Hamiltonian

H4 in Ref. [86] with spin-1/2) numerically appears to exhibit strong fracture within

several symmetry sectors. However, the addition of longer-range CoM preserving

terms numerically appears to cause the Hilbert space to fracture only weakly [86],

with the fracture disappearing with the addition of infinite-range CoM preserving

terms, even if the interaction strength decays exponentially with range [228].

By definition, distinct Krylov subspaces are dynamically disconnected i.e., no

state initialized completely within one of the Krylov subspaces can evolve out to

a different Krylov subspace. Indeed, exponentially many of these Krylov sub-

spaces are one-dimensional static configurations—product states that are eigen-

states of H. For instance, the Hamiltonian vanishes on any product state that

does not contain the patterns “· · · 0110 · · · ” or “· · · 1001 · · · ”, since those are the

only configurations on which terms of H act non-trivially (see Eq. (5.2)). The

charge-density-wave (CDW) state

|1111000011110000 . . . . . . 1111000011110000〉

is one example of a static configuration that is an eigenstate. In terms of the

180

composite degrees of freedom we can equivalently consider configurations with

only +, −, and no spins, such as

|· · ·++−−++−− · · · 〉 ,

with a pattern that alternates between + and − with ‘domain walls’ that are

at least 2 sites apart. According to Eqs. (5.8)-(5.12), all terms of the Hamil-

tonian vanish on these configurations: since there are exponentially many such

patterns, there are equally many one-dimensional Krylov subspaces. We can also

construct small Krylov subspaces by embedding finite non-trivial blocks, on which

the Hamiltonian acts non-trivially, into the static configurations, thereby leading

to exponentially many Krylov subspaces of every size [86, 87]. For example, the

following configurations |ψ±〉

|ψ±〉 = 1√2(|++−− · · ·++−− ↑↓ ++−− · · ·++−−〉

± |++−− · · ·++−− ↓↑ ++−− · · ·++−−〉) (5.21)

are composed of one non-trivial block ↑↓ sandwiched within a frozen configuration,

and they thus have energies E± = ±1. Exponentially many configurations with

energies E = ±1 can be constructed by changing the frozen configuration around

the non-trivial block.

The presence of exponentially many static states (within each symmetry sec-

tor) in the the Hilbert space leaves an imprint on the dynamical behaviour of

such systems. Specifically, time-evolution starting from randomly chosen product

states looks highly non-generic from the perspective of the full Hilbert space. For

181

example, in the absence of Krylov fracture one typically expects that the bipar-

tite entanglement entropy evolves to the Page value [107], the average bipartite

entanglement entropy of states in the Hilbert space. For a system of Hilbert

space dimension D [L] = 2L, the Page value is logD [L/2] ≈ L/2 log 2. However,

in the presence of Krylov fracture, we expect that the late-time bipartite entan-

glement entropy of product states |ψ0〉 is smaller and typically ∼ logDK [L/2],

where DK [L] is the dimension of the Krylov subspace K (H, |ψ0〉) for a system

size L. The phenomenon of Krylov fracture can thus be regarded as a breaking

of ergodicity with respect to the full Hilbert space, resulting in (at the very least)

violation of strong ETH.

However, what remains unclear is whether, for systems exhibiting Krylov frac-

ture, thermalization occurs within each of the Krylov subspaces. Of course, ther-

malization or ETH-violation are only well-posed concepts for large Krylov sub-

spaces K (with dimension DK[L] → ∞ as L → ∞)† and do not have a clear

meaning when the Krylov subspace has a finite dimension in the thermodynamic

limit, as is the case for the exponentially many static configurations discussed

above. Indeed, there exist exponentially large Krylov subspaces of the Hamilto-

nian Eq. (5.1) at filling ν = p/(2p+1) for which Krylov-restricted thermalization

appears to hold for most initial states, as recently demonstrated by some of the

present authors [78]. There, we demonstrated the existence of Krylov subspaces

with Wigner-Dyson level statistics, despite such Krylov subspaces hosting quan-

tum scars i.e., evenly spaced towers of anomalous states in the spectrum that lead

to revivals in the fidelity of time evolution from particular initial states. Those†Note that the dimension of the Krylov subspace DK[L] could in principle scale polynomially

with L; however, we are not aware of any such example in the pair-hopping model Eq. (5.1).

182

Subspace Root Quantum Numbers HamiltonianSpin |↑↓ · · · ↓↑〉 N↑ HXX [N ]

Single +− |↑ · · · ↓ +− ↑ · · · ↓〉 N(1)↑ , N

(2)↑ HXX [N − 1]

2 separated +− |+− ↑ · · · ↓ +−〉 N(1)↑ , N

(2)↑ ≥ 1, N

(3)↑ HXX [N − 2]

2 adjacent +− |↑ · · · ↓ +−+− ↑ · · · ↓〉 N(1)↑ , N

(2)↑ = 0, N

(3)↑ HXX [N − 1]

X separated +− |+− ↑ · · ·+− · · · ↓ +−〉 N(1)↑ , N (j)

↑ ≥ 1, N(X)↑ HXX [N − (X − 1)]

X adjacent +− |· · ·+− · · ·+− · · · 〉 N(1)↑ , N (j)

↑ = 0, N(X)↑ HXX [N − 1]

Table 5.1: Table of integrable Krylov subspaces (by no means an exhaustive list)of the pair-hopping model for system size L = 2N , with OBC at half-filling. Foreach type of Krylov subspace, we provide the root configuration generating it, theassociated quantum numbers, and the Hamiltonian restricted to that subspace.Dipole subspaces for the oppositely oriented −+ dipoles can be constructed anal-ogously (see main text for discussion).

Krylov subspaces are examples of ones that violated Krylov-restricted strong ETH,

although Krylov-restricted weak ETH is satisfied. However, it has not yet been

established if Krylov-restricted weak ETH is necessarily satisfied for large dimen-

sional Krylov subspaces, or if there are examples of semi-integrable systems with

both integrable and non-integrable Krylov subspaces, opening the door to further

violations of ergodicity within Krylov sectors.

Thus, in what follows we will focus on high dimensional irreducible Krylov sub-

spaces K (H, |ψ〉), defined as those with exponentially large dimension DK [L] ∼

αL as L→∞ (α > 1), and which satisfy

K (H, |ψ〉) 6= K (H, |ψ1〉)⊕K (H, |ψ2〉) (5.22)

for any product states |ψ1〉 and |ψ2〉, after resolving charge and center-of-mass

symmetries. Remarkably, we find several examples of both integrable and non-

integrable subspaces in the model Eq. (5.1), demonstrating the rich dynamical

183

structure inherent in systems with fractured Hilbert spaces. Studying the dynam-

ics of root states that generate large irreducible Krylov subspaces thus allows us

to establish that integrability or non-integrability of a system is correctly defined

only within each Krylov subspace.

5.5 Integrable subspaces

In this section, we illustrate several integrable irreducible Krylov subspaces with

exponentially large dimension present in the pair-hopping model Eq. (5.1).

5.5.1 Spin subspace

The simplest example of a large integrable Krylov subspace can be generated

by a root state |ψ0〉 (see Eq. (5.18)) which is any product state of only spin

degrees of freedom: ↑ and ↓ as defined in Eq. (5.7). From Eq. (5.8), we find that

the Hamiltonian restricted to this subspace can be written as a nearest neighbor

Hamiltonian with actions:

|↑↑〉 → 0, |↓↓〉 → 0, |↑↓〉 ↔ |↓↑〉 , (5.23)

where |a〉 → 0 and |a〉 ↔ |b〉 represent the action of a single term of the Hamilto-

nian. Thus, starting from a root state with N↑ spin ↑’s (and hence (N −N↑) spin

↓’s), such as

|↑↓↑↑↓〉 , (N,N↑) = (5, 3),

the action of the Hamiltonian only rearranges the spins.

In particular, note that: (i) The number of ↑’s and ↓’s in the root state N↑

184

and N −N↑ respectively are preserved upon the action of the Hamiltonian, (ii) no

fractons (i.e. +’s or −’s) are created, and (iii) all product configurations with N

spins and a fixed value of N↑ are part of the Krylov subspace K (H, |ψ0〉) associated

with the root state |ψ0〉. Furthermore, since the Hamiltonian restricted to this

subspace only interchanges the spins (see Eq. (5.23)), it maps exactly onto that

of the spin-1/2 XX model:

HXX [N ] ≡N∑j=1

(σ+j σ

−j+1 + σ−

j σ+j+1

), (5.24)

where σ+j and σ−

j are onsite Pauli matrices. This mapping was first noted

in earlier works on half-filled Landau levels [215, 216, 229]. As is well known, the

Hamiltonian Eq. (5.24) can be solved using a Jordan-Wigner transformation [230],

upon which it maps onto a non-interacting problem. We numerically observe that

the full ground state of the Hamiltonian Eq. (5.1) belongs this Krylov subspace

with (N,N↑) =(N,⌊N2

⌋).

An important note regarding symmetries: each Krylov subspace generated from

a root state with only spins and with a fixed N↑ (dubbed the spin Krylov sub-

space) only generates one symmetry sector of the XX model with a fixed Sz. All

symmetry sectors of the XX model can be generated by starting from root states

with different N↑, so that the full spectrum of the XX model of N sites is embed-

ded within the spectrum of the pair-hopping Hamiltonian H (5.1), both for OBC

and PBC.

With respect to the symmetries of H, these Krylov subspaces lie within the

sector (Q,D, Sz) = (0, 0, 2N↑ − N), where Q, D, and Sz are the total charge,

dipole moment, and spin respectively, discussed in Sec. 5.3.2. However, these are

185

not the only states within that (Q,D, Sz) symmetry sector, providing evidence for

the Krylov fracture in the pair-hopping Hamiltonian H. For example, the product

state

|∗ · · · ∗+−−+ ∗ · · · ∗〉 , (5.25)

where ∗ = ↑, ↓ and with (N↑ − 1) ↑’s (and hence (N − N↑ − 1) ↓’s) lies within

the symmetry sector (Q,D, Sz) = (0, 0, 2N↑ − N) but outside the spin Krylov

subspace constructed above.

5.5.2 Single dipole subspace

Restricting our attention to OBC, we now demonstrate the existence of another

set of integrable Krylov subspaces K (H, |ψ0〉), which are generated from root

states containing only a single dipole. Such root states are of the form

|ψ0〉 = |∗ · · · ∗+− ∗ · · · ∗〉 , |ψ0〉 = |∗ · · · ∗ −+ ∗ · · · ∗〉 , (5.26)

where ∗ = ↑, ↓. The action of the Hamiltonian Eq. (5.1) on configurations of the

form Eq. (5.26) is given by

|↓ +−〉 ↔ |+− ↓〉 , |↑ −+〉 ↔ |−+ ↑〉 |↑ +−〉 → 0, |+− ↑〉 → 0

|↓ −+〉 → 0, |−+ ↓〉 → 0. (5.27)

Since dipole moment is conserved, the dipole does not “disintegrate” under the

action of the Hamiltonian Eq. (5.27), i.e. the dipole does not separate into its

constituent + and − fractons. As it turns out, Krylov subspaces generated by

root states of the form (5.26) with N sites are isomorphic to Hilbert spaces of

186

(N −1) spin-1/2’s, with the effective Hamiltonians within these Krylov subspaces

given by XX models of (N − 1) sites. In the following, we focus on the Krylov

subspace corresponding to a +− dipole. As we discuss later, the generalization

to −+ dipoles follows similarly.

To show this, we first observe that as a consequence of Eq. (5.27), a dipole +−

in the root state can never cross an ↑ spin to its left or to its right. In other

words, the dipole +− can only hop left (right) if there is a ↓ spin immediately to

its left (right). Hence, all product states in the Krylov subspace generated by a

root state |ψ0〉 with one dipole +− preserve the number of ↑ spins to the left and

right of the dipole separately. Denoting these conserved quantities by N(1)↑ and

N(2)↑ respectively, we see that product states in the Krylov subspace K (H, |ψ0〉)

always have the form

|∗ · · · ∗+− ∗ · · · ∗〉︸ ︷︷ ︸N

(1)↑

︸ ︷︷ ︸N

(2)↑

, (5.28)

where ∗ = ↑, ↓. This Krylov subspace can thus be uniquely labelled by the tuple

(N,N(1)↑ , N

(2)↑ ). For example, the Krylov subspace K (H, |ψ0〉) generated by the

configuration |ψ0〉 = |↑↓ +− ↑↓〉 with OBC consists of the following basis states:

|↑↓ +− ↑↓〉 , |↓↑ +− ↑↓〉 , |↑↓ +− ↓↑〉 , |↓↑ +− ↓↑〉

|↑ +− ↓↑↓〉 , |↑ +− ↑↓↓〉 , |↑ +− ↓↓↑〉 , |↑↓↓ +− ↑〉 , |↓↑↓ +− ↑〉 , |↓↓↑ +− ↑〉 .(5.29)

Note that all the states in K (H, |ψ0〉) are labelled by (N,N(1)↑ , N

(2)↑ ) = (6, 1, 1).

In order to map configurations of the form Eq. (5.28) onto an effective spin-1/2

Hilbert space, note that the rules of Eq. (5.27) are identical to those of Eq. (5.8)

when the dipole +− is replaced by an ↑ spin. This observation allows us to

187

establish two crucial results on the single-dipole Krylov subspace K (H, |ψ0〉).

Firstly, product states in the single dipole Krylov subspace consisting of a +−

dipole can be uniquely mapped onto product states of (N − 1) spin-1/2’s with

(N(1)↑ + N

(2)↑ + 1) ↑’s by replacing the +− dipole with an ↑. For example, the

following holds:

|↑↑↓ +− ↑↓↑↑↑〉(A)

⇐⇒ |↑↑↓↑↑↓↑↑↑〉(B)

, (5.30)

where configuration (A) in the Krylov subspace with (N,N(1)↑ , N

(2)↑ ) = (10, 2, 4)

maps onto the configuration (B) in the spin subspace with (N,N↑) = (9, 6) by

replacing the +− dipole with an ↑. The inverse mapping from the spin-1/2 Hilbert

space of (N−1) sites and (N(1)↑ +N

(2)↑ +1) ↑’s to the single dipole Krylov subspace

(N,N(1)↑ , N

(2)↑ ) proceeds by identifying one ↑ to be the +− dipole such that the

resulting configuration has the correct N (1)↑ and N

(2)↑ . For instance in Eq. (5.30),

given (N,N(1)↑ , N

(2)↑ ) = (10, 2, 4), the mapping from (B) to (A) is possible only if

the third ↑ in the configuration (B) is replaced by a +− dipole.

The mapping for the single −+ dipole subspace follows analogously, with ↑

replaced by ↓ i.e., by identifying −+’s with ↓’s instead. In that case, the quantities

N(1)↓ and N

(2)↓ , defined as

|∗ · · · ∗ −+ ∗ · · · ∗〉︸ ︷︷ ︸N

(1)↓

︸ ︷︷ ︸N

(2)↓

, (5.31)

are preserved within the Krylov subspace. Thus, the single dipole Krylov subspace

with OBC and a fixed (N,N(1)↑ , N

(2)↑ ) (resp. (N,N

(1)↓ , N

(2)↓ )) is isomorphic to the

Hilbert space of (N−1) spin-1/2’s with (N(1)↑ +N

(2)↑ +1) ↑’s (resp. (N (1)

↓ +N(2)↓ +1)

↓’s). Secondly, since Eq. (5.27) is identical to Eq. (5.23) when the dipole +− (resp.

−+) is replaced with an ↑ (resp. ↓), the effective Hamiltonian within each such

188

Krylov subspace is the XX model of (N − 1) sites with OBC.‡ In particular, the

spectrum of H in Eq. (5.24) restricted to the single Krylov subspace labelled

by (N,N(1)↑ , N

(2)↑ ) (resp. (N,N (1)

↑ , N(2)↑ )) is precisely the spectrum of the quantum

number sector Sz = (2(N(1)↑ +N

(2)↑ )+3−N) (resp. Sz = −(2(N (1)

↓ +N(2)↓ )+3−N))

of the XX model.

Note that with PBC this Krylov subspace is no longer isomorphic to the spin-

1/2 Hilbert space of the XX model, since the inverse mapping from the spin-1/2

Hilbert space to the dipole subspace is not unique. Thus, the effective Hamiltonian

within this Krylov subspace cannot map exactly onto the XX model of Eq. (5.24)

with PBC, and it remains unclear whether or not the resulting Hamiltonian is

integrable for any finite system size.

5.5.3 Multidipole subspaces

We now consider Krylov subspaces generated by root configurations containing

multiple identically oriented dipoles. All such subspaces turn out to be integrable

and governed by effective XX Hamiltonians of various sizes. As with a single

dipole discussed in the previous section, spins and dipoles interact according to

Eq. (5.27). A crucial property of these rules, which we will make use of throughout

this section, is that the +− (resp. −+) dipole cannot cross any ↑ (resp. ↓) spins

under the action of the Hamiltonian H.

We first illustrate the case where the root state contains two +− dipoles before

discussing the general setting. Since the dipoles +− cannot cross ↑’s, the Krylov

subspace generated from a root state with two identically oriented dipoles pre-‡Once an ↑ spin is identified, note that the action of the XX Hamiltonian also preserves N (1)

↑

and N (2)↑ , the number of ↑ spins to the left and to the right of the identified ↑ spin respectively.

189

serves three quantities of the root state: (N (1)↑ , N

(2)↑ , N

(3)↑ ), depicted schematically

by the following configurations:

|∗ · · · ∗+− ∗ · · · ∗+− ∗ · · · ∗〉︸ ︷︷ ︸N

(1)↑

︸ ︷︷ ︸N

(2)↑

︸ ︷︷ ︸N

(3)↑

, (5.32)

where ∗ = ↑, ↓. That is, for a Krylov subspace generated by root states with two

+− dipoles, the number of ↑ spins to the left of the left dipole, in between the two

dipoles, and to the right of the right dipole are each separately conserved. Thus,

the quantities (N,N(1)↑ , N

(2)↑ , N

(3)↑ ) uniquely label the Krylov subspace.

We now restrict our discussion to the Krylov subspace containing two +−

dipoles, with the generalization to the two −+ dipole subspace being straight-

forward. Provided N(2)↑ ≥ 1 in the root state |ψ0〉, the two dipoles are always

separated by an ↑ spin and can never be adjacent to each other; the action of the

Hamiltonian is therefore entirely specified by Eq. (5.27). Product states in the

Krylov subspace can be mapped onto configurations of (N − 2) spin-1/2’s with(N

(1)↑ +N

(2)↑ +N

(3)↑ + 2

)↑’s by replacing the +− dipoles by ↑’s. For example,

|↑↓↑ +− ↑↓↑ +− ↑↓〉(A)

⇐⇒ |↑↓↑↑↑↓↑↑↑↓〉(B)

, (5.33)

where the configuration (A) in the two-dipole Krylov subspace labeled by(N,N

(1)↑ , N

(2)↑ , N

(3)↑

)=

(12, 2, 2, 1), maps onto configuration (B).

Similar to the single dipole case, the inverse mapping is unique once (N,N (1)↑ , N

(2)↑ , N

(3)↑ )

are specified. This inverse mapping proceeds by identifying two of the ↑’s to be

+− dipoles such that the resulting configuration has the required values of N (1)↑ ,

190

N(2)↑ , and N

(3)↑ . For example, given that (N,N

(1)↑ , N

(2)↑ , N

(3)↑ ) = (12, 2, 2, 1), the

two-dipole configuration (A) in Eq. (5.33) is the unique two-dipole configuration

corresponding to spin configuration (B).

The mapping for the two-dipole subspace with −+ dipoles follows analogously,

with ↑ replaced by ↓ i.e., by identifying −+’s with ↓’s instead. The action of

the Hamiltonian is completely specified by Eq. (5.27) when the dipoles are not

allowed to be adjacent each other; as discussed in Sec. 5.5.2, Eq. (5.27) is identical

to Eq. (5.23) when the +− (resp. −+) dipole is identified with ↑ (resp. ↓)

spin. Thus, the Hamiltonian restricted to the two +− (resp. −+) dipole Krylov

subspace is identical to the XX model of (N − 2) sites within the Sz = (2(N(1)↑ +

N(2)↑ +N

(3)↑ ) + 6−N) (resp. Sz = −(2(N (1)

↓ +N(2)↓ +N

(3)↓ ) + 6−N)) sector.

We emphasize that the two-dipole Krylov subspace of N is isomorphic to the

spin-1/2 Hilbert space of (N − 2) sites only when the two +− (resp. −+) dipoles

have at least one ↑ (resp. ↓) spin between them i.e., only if N (2)↑ ≥ 1 (resp.

N(2)↓ ≥ 1). When the two dipoles are adjacent to each other, using Eqs. (5.11)

and (5.12) we find that the action of the Hamiltonian H reads

|+−+−〉 ↔ |↓ + ↑ −〉 , |+−+−〉 ↔ |+ ↑ − ↓〉 ,

|−+−+〉 ↔ |↑ − ↓ +〉 , |−+−+〉 ↔ |− ↓ + ↑〉 . (5.34)

As a consequence, the action of the Hamiltonian on root states of the form

|· · ·+−+− · · · 〉 result in the “disintegration” of dipoles, resulting in configu-

rations of the form:

|· · · ↓ + ↑ − · · · 〉 , |· · ·+ ↑ − ↓ · · · 〉 ,

191

which cannot be mapped onto a configuration of (N − 2) spin-1/2’s through the

map described earlier in this section. Nevertheless, we find that such Krylov

subspaces does map onto the XX model, albeit one with (N − 1) spin-1/2’s [88].

The preceding discussion straightforward generalizes to three or more dipoles.

For a Krylov subspace generated by a root state containing n identically oriented

dipoles, with OBC the system can be partitioned into (n+1) segments separated

by the dipoles. We introduce the quantities N (1)↑ , N

(2)↑ , · · · , N (n+1)

↑ , where N (j)↑

(resp. N (j)↓ ) represents the number of ↑ (resp. ↓) spins in the j-th segment of the

chain in the root state:

1 2 n−1 n

|· · ·+− · · ·+− · · ·+− · · ·+− · · · 〉︸︷︷︸N

(1)↑

︸︷︷︸N

(2)↑

︸︷︷︸N

(n)↑

︸︷︷︸N

(n+1)↑

, (5.35)

with the superscripts 1, 2, · · · , n indexing the dipoles. Since a +− dipole is not

allowed to cross an ↑ spin under the action of the Hamiltonian, the quantities

N (j)↑ ≥ 1 are invariant under the dynamics i.e., these quantities are identical for

all product states within the Krylov subspace generated by the root state of the

form Eq. (5.35). As with two dipoles, this is true provided no dipoles are adjacent

in the root state, which corresponds to the constraint N (j)↑ ≥ 1 for any j.

In this case (N (j) 6= 0 ∀ j), the n dipole Krylov subspace exactly maps onto a

spin-1/2 Hilbert space with (N−n) sites and (n+n+1∑j=1

N(j)↑ ) ↑’s by identifying each

+− dipole with an ↑ spin. For example,

|↑↓ +− ↓↑↑ +− ↓↑ +− ↑↑〉(A)

⇐⇒ |↑↓↑↓↑↑↑↓↑↑↑↑〉(B)

, (5.36)

192

where n = 3 and where the three dipole configuration (A) with (N(1)↑ , N

(2)↑ , N

(3)↑ , N

(4)↑ ) =

(1, 2, 1, 2) maps onto the spin configuration (B). This mapping onto the spin-1/2

Hilbert space is invertible provided the tuple (N(1)↑ , N

(2)↑ , · · · , N (n+1)

↑ ) is known,

and it proceeds by identifying n ↑ spins in each product configuration with +−

dipoles such that the resulting configuration has the requisite (N (1)↑ , N

(2)↑ , · · · , N (n+1)

↑ )

values. For example, given (N(1)↑ , N

(2)↑ , N

(3)↑ , N

(4)↑ ) = (1, 2, 1, 2), configuration (B)

in Eq. (5.36) uniquely maps onto (A) by identifying the appropriate ↑ spins with

+− dipoles.

The mapping with −+ dipoles proceeds in a similar way by replacing the −+

dipole by ↓. The quantities N (j)↓ are thus preserved within the Krylov subspaces,

where1 2 n−1 n

|· · · −+ · · · −+ · · · −+ · · · −+ · · · 〉︸︷︷︸N

(1)↓

︸︷︷︸N

(2)↓

︸︷︷︸N

(n)↓

︸︷︷︸N

(n+1)↓

. (5.37)

Since the Hamiltonian Eq. (5.27) is identical to Eq. (5.8) upon the identification

of dipoles with spins, the Hamiltonian restricted to the Krylov subspace for the n

+− (resp. −+) dipole case is the XX model with (N−n) sites within the quantum

number sector Sz = (3n+n+1∑j=1

N(j)↑ −N) (resp. Sz = −(3n+

n+1∑j=1

N(j)↓ −N)).

When N (j)↑ = 0 or N (j)

↓ = 0 for some j in the root state Eq. (5.35), the mapping

prescribed above fails because the action of the Hamiltonian causes the adjacent

dipoles to disintegrate, as shown in Eq. (5.34). Nevertheless, we find that the

Krylov subspace remains integrable even if some dipoles in the root state are

adjacent. Specifically, we find that the Hamiltonian restricted to a Krylov sub-

space with only n +− (resp. −+) dipoles is the XX model of (N − n+X) sites,

where X is the number of segments j containing no spins, such that N (j)↑ = 0

193

(resp. N(j)↓ = 0). For example, the effective Hamiltonians restricted to the

Krylov subspaces generated by the root states |∗ · · · ∗+−+−+− ∗ · · · ∗〉 and

|∗ · · · ∗+−+− ∗ · · · ∗+− ∗ · · · ∗〉 , where ∗ = ↑, ↓ are the XX models acting on

(N − 1) and (N − 2) spin-1/2’s respectively.

As was the case for a single dipole, the mapping onto XX models does not

work with PBC. However, it is not clear if the effective Hamiltonian restricted

to this sector with PBC is solvable for a finite system size, although integrabil-

ity of this sector should be restored in the thermodynamic limit and the energy

spectrum should display Poisson level statistics for a large enough system size.

Finally, we note that upon the addition of electrostatic terms or disorder, the

spin subspace described in Sec. 5.5.1 maps onto the XXZ model or disordered XX

model, and thus remains integrable. However, the dipole subspaces are no longer

integrable, and they show all the signs of usual non-integrability, including GOE

level statistics [103].

5.5.4 Systematic construction of integrable subspaces

Having illustrated the existence of several integrable Krylov subspaces of the pair-

hopping model Eq. (5.1), we briefly discuss a general prescription for constructing

additional irreducible integrable subspaces by using the integrable subspaces of

Secs. 5.5.1-5.5.3 as building blocks. As also emphasized in Refs. [86, 87], one can

introduce blockades i.e., regions of the chain on which terms of the Hamiltonian

vanish. For example, consider the following root state with a configuration of the

form:

|∗ · · · ∗++ · · ·++ ∗ · · · ∗〉︸ ︷︷ ︸A

︸ ︷︷ ︸B

, (5.38)

194

where ∗ = ↑, ↓, with N+ ≥ 2 and N− = 0. Following the rules Eqs. (5.8)-(5.12),

the Hamiltonian can act non-trivially only on sites contained within regions A

and B of the root state Eq. (5.38).

Due to this, all basis states of the Krylov subspace generated from the root

state Eq. (5.38) retain the same schematic form, with ++ · · ·++ (N+ ≥ 2) acting

as a blockade that spatially disconnects two parts of the Krylov subspace.

Thus, one can show that the effective Hamiltonian restricted to such blockaded

Krylov subspaces is simply given by the sum of two independent XX models acting

on distinct degrees of freedom lying in regions A and B. Note that blockades can

also be constructed using exponentially many other “static” patterns [86,87], such

as − − · · · − −, + + − − · · · + + − −, or ++ ↑ · · · ↑ ++, which in turn lead to

exponentially many integrable subspaces.

Similarly, we can also introduce blockades for the dipole Krylov subspaces con-

sidered in Secs. 5.5.2-5.5.3, as long as the dipoles do not interact with the blockade.

For example, consider the root configuration of the form of Eq. (5.38) where region

A is a root configuration for an integrable subspace with one or more −+ dipoles,

and region B is a root configuration for an integrable subspace with +− dipoles:

|∗ · · · ∗ −+ ∗ · · · ∗++ · · ·++ ∗ · · · ∗+− ∗ · · · ∗〉︸ ︷︷ ︸A

︸ ︷︷ ︸B

, (5.39)

where ∗ =↑, ↓. Upon successive applications of the Hamiltonian on the root state

of Eq. (5.39), the dipoles in regions A and B do not interact with the string of

+’s in between the regions. Thus, the string of +’s acts as a blockade, and the

Krylov subspaces generated by such root configurations are integrable, since the

restricted Hamiltonian is a sum of XX models on regions A and B. While we

195

have only illustrated the simplest cases where blockades are introduced between

regions A and B, each of which are integrable regions that do not interact with

the blockade, we can of course generalise by introducing n blockades separating

n+1 regions, each of which contain the integrable subspaces that do not interact

with the neighboring blockades. In such a case, the Hamiltonian restricted to the

Krylov subspace is a sum of n+ 1 independent XX models.

A detailed study delineating all integrable subspaces of the pair-hopping model

Eq. (5.1) is beyond the scope of this work. Nevertheless, the above examples suf-

fice to illustrate the existence of exponentially many integrable Krylov subspaces,

clearly establishing the possibility of emergent, Krylov-restricted integrability in

systems exhibiting Krylov fracture.

5.6 Non-integrable subspaces and Krylov-Restricted ETH

Given that large swaths of the spectrum of the pair-hopping Hamiltonian are solv-

able, it is natural to ask whether this model is completely integrable. The standard

diagnostic for probing non-integrability of some Hamiltonian is the appearance of

random matrix behavior within a sector resolved by symmetries of that Hamilto-

nian. For example, the energy level statistics [38,103] and the matrix elements of

local operators in the energy eigenbasis (according to ETH) [15] are expected to

follow random matrix behavior for non-integrable systems.

Generally, in unconstrained models, symmetry sectors are themselves examples

of well-defined dynamically disconnected Krylov subspaces. In other words, a

root-state which is an eigenstate of the symmetry typically generates a Krylov

subspace which spans all states within that symmetry sector. However, for systems

196

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

∆E

0.0

0.2

0.4

0.6

0.8

1.0

P(∆E

)

(a) GOEPoisson

0.4 0.2 0.0 0.2 0.4

E/N

0.2

0.0

0.2

0.4

0.6

0.8

⟨ E|QN/2|E⟩ −⟨ Q

N/2⟩ E

(b) N = 12N = 14N = 16N = 18

102 103 10410-2

10-1

100

Figure 5.1: (a) Level statistics within the Krylov subspace K (H, |ψ0〉) generatedby various root states |ψ0〉 with OBC. Red: |ψ0〉 = |↓↑↑↑↓↓↓ −++− ↓↓↓↑↑↓↓〉,Blue: |ψ0〉 = |↑↓↑↓↑ + ↑↑↑↑↓↓↓ − ↓↑↓↑↓〉. The configurations of spins in the rootconfigurations have been chosen to ensure that the Krylov subspace does not haveany symmetries. The standard 〈r〉 parameter [105] in these subspaces is 0.5331 and0.5276 respectively, close to the GOE value of 0.53. (b) Evidence for ETH in thenon-integrable Krylov subspace K (H, |ψ0〉) generated by the root state shown inEq. (5.42), which for N = 18 reads |ψ0〉 = |↑↓↑↓↑↓↑ −++− ↓↑↓↑↓↑↓〉. CouplingsJj of the terms are chosen from a uniform distribution [1 −W, 1 + W ], withW = 0.1 in order to break symmetries but preserve the Krylov fractured structureof the Hilbert space. Main: The difference between the eigenstate and thermalexpectation values of the charge operator at energy E. Inset: The standarddeviations of that difference as a function of the Hilbert dimension DK scales∼ 1/

√DK (dotted line) for two operators: the charge operator QN/2 (blue) and

the spin operator SzN/2 (red), consistent with ETH within the Krylov subspaceK (H, |ψ0〉).

exhibiting Krylov fracture, there exist several dynamically disconnected Krylov

subspaces within each symmetry sector. As was also emphasized by Refs. [86,87],

resolving eigenstates by symmetries alone may hence be insufficient for identifying

ergodicity, given the possibility of Krylov fracture.

Thus, we pose the crucial question that motivates the title of the paper: Whether

197

symmetries are only a subset of the more general phenomena of Krylov fracture,

and if ergodicity or its absence should correspondingly be defined within dynam-

ically disconnected irreducible Krylov subspaces. In the previous section, we en-

countered examples of Krylov subspaces within symmetry sectors which display

the characteristic trademarks of integrable systems e.g., Poisson level statistics.

Now, we wish to ask whether Krylov subspaces that are not integrable exhibit con-

ventional diagonostics of ergodic systems, such as Wigner-Dyson level statistics

and ETH [15]. Of course, random matrix theory is a statement about “large” ma-

trices i.e., in the limit that the size of the matrix goes to infinity; consequently, the

question of thermalization within Krylov subspaces is only well-posed for “large”

Krylov subspaces, whose size tends to infinity in the thermodynamic limit. Thus,

we explore some simple non-integrable Krylov subspaces of the pair-hopping model

Eq. (5.1) and, in the process, establish the notion of Krylov-restricted ETH.

Indeed, there exist Krylov subspaces of the pair-hopping Hamiltonian which are

not integrable. Consider for instance the Krylov subspace generated by the root

state containing both +− and −+ dipoles:

|ψ0〉 = |∗ · · · ∗ −++− ∗ · · · ∗〉 , (5.40)

where ∗ = ↑, ↓. Since the dipoles are of opposite orientation, the mapping of

the +− and −+ dipoles to ↑ and ↓ spins would only be justified if |−++−〉 ↔

|+−−+〉 under the action of the Hamiltonian, which is strictly prohibited by the

rules given in Eqs. (5.8)-(5.12). As a result, the Hamiltonian restricted to this

Krylov subspace does not need to map onto an integrable model. Another exam-

ple is the Krylov subspace generated by the root state containing two separated

198

fractons:

|ψ0〉 = |∗ · · · ∗+ ∗ · · · ∗ − ∗ · · · ∗〉 , (5.41)

where ∗ = ↑, ↓, and the ∗ · · · ∗ in between the + and − contains both ↑ and ↓

spins. The latter condition is required to ensure that |ψ0〉 does not belong to any

of the integrable multidipole Krylov subspaces.

We have numerically studied the behaviour of Krylov subspaces generated by

root states such as those given in Eqs. (5.40) and (5.41). As shown in Fig. 5.1(a),

we find that eigenstates of the Hamiltonian within these Krylov subspacesK (H, |ψ0〉)

exhibit GOE level statistics, providing evidence for the non-integrability of the

Krylov subspace. We further conjecture that such non-integrable Krylov sub-

spaces satisfy the Eigenstate Thermalization Hypothesis (ETH) [14–18].

Here, we want to test whether Eq. (12) holds within a non-integrable Krylov

subspace. We focus on the Krylov subspace with the root states (with OBC):

|ψ0〉 =

|↑↓ · · · ↑↓ −++− ↑↓ · · · ↑↓〉 if N = 4p

|↑↓ · · · ↑↓↑ −++− ↓↑↓ · · · ↑↓〉 if N = 4p+ 2, (5.42)

with two dipoles −+ and +− placed at the center of the chain. Furthermore,

to probe the validity of Eq. (12), we need to choose an operator O that pre-

serves the Krylov subspaces. Hence we choose the charge operator on the N/2-

th site O = QN/2, which is diagonal in the basis of product states. Since the

Krylov subspace K (H, |ψ0〉) has symmetries (e.g. inversion symmetry), we add

disorder to the couplings of the pair-hopping Hamiltonian, which does not af-

fect the structure of the Krylov subspaces of the Hamiltonian, and focus on

testing the ergodicity within the Krylov subspace. To probe the validity of

199

Eq. (12) within non-integrable Krylov subspaces, in Fig. 5.1(b) we plot the quan-

tity(〈E|O|E〉 − O (E)

), as a function of E, where |E〉 is the eigenstate with

energy E. The inset show the variance of the difference as a function of the

Krylov subspace dimension DK.

Two observations in Fig. 5.1(b) suggest the validity of ETH within the Krylov

subspace. Firstly, the quantity 〈E|O|E〉− O (E) is centered about 0, which shows

that eigenstate expectation values approach the thermal expectation value. Sec-

ondly, the standard deviation of the difference (shown in the inset) scales as

∼ 1/√DK, the dimension of the Krylov subspace. Hence these observations pro-

vide evidence for “diagonal ETH” within non-integrable Krylov subspaces, sup-

porting the existence of Krylov-restricted ETH in systems exhibiting Krylov frac-

ture.

5.7 Quasilocalization from Thermalization

Based on the results of the previous section, which established the phenomenon of

Krylov-restricted ETH, we expect that the long-time behaviour of typical states

within a particular non-integrable (resp. integrable) Krylov subspace coincides

with the Gibbs ensemble (resp. generalized Gibbs ensemble) restricted to that

subspace. Such Krylov-restricted thermalization can lead to surprising behaviour

within some Krylov subspaces. For example, in the following we show that the

thermal expectation value of charge density on the chain within a particular Krylov

subspace is spatially non-uniform for any finite system size.

To illustrate this behaviour, we consider the dynamics of a single fracton im-

mersed in a spin background i.e., we study the Krylov subspace generated by the

200

0 20 40 60 80 100

T

0.0

0.2

0.4

0.6

0.8

1.0

⟨ ψ 0|eiHTQxe−

iHT|ψ

0

⟩

(a) x = 8x = 2

2 4 6 8 10 12 14

x

0.00

0.05

0.10

0.15

0.20

⟨ Q x⟩ α

(b) α = Infinite-temperatureα = Late-time

Figure 5.2: (Color online) (a) Time-evolution of the expectation value of on-sitecharge operators for the middle site and a site away from the middle under thepair hopping Hamiltonian with PBC, starting from an initial state of the form|ψ0〉 = |∗ · · · ∗+ ∗ · · · ∗〉 where ∗ = ↑, ↓ for N = 15 with total spin Sz = 0.The horizontal lines show the infinite-temperature expectation values of the samecharge operators. Data averaged over 10 configurations of the ∗’s such that Sz = 0.(b) Late-time charge profile on sites of the chain matches the infinite temperaturevalue within the Krylov subspace K (H, |ψ0〉). They both show a peak on themiddle site, providing an example of quasilocalization from thermalization.

root state with PBC:

|ψ0〉 = |∗ · · · ∗+ ∗ · · · ∗〉 , (5.43)

where ∗ =↑, ↓ such that N↑ = N↓. The configuration |ψ0〉 thus belongs to the

quantum number sector Q = 1, D = exp (iπ(N + 1)/N) , Sz = 0, where N is the

length of the chain. Since we impose PBC here, all configurations of ∗’s in the

root state generate the same Krylov subspace as the spins can rearrange amongst

themselves under the action of the Hamiltonian (see Eq. (5.8)). Hence, in the fol-

lowing, we only explicitly describe the action of the Hamiltonian on the fractons,

given that all possible spin configurations (with N↑ = N↓) are generated within

201

this subspace. There are two possibilities for how the state |ψ0〉 of Eq. (5.43)

evolves under one application of the Hamiltonian H: either the spins can rear-

range amongst themselves or the fracton moves by emitting a dipole, according

to Eq. (5.11). Since we are only focusing on the fracton, in the latter case, the

new basis state reads

|ψ1〉 = |∗ · · · ∗+−+ ∗ · · · ∗〉 , (5.44)

where ∗ = ↑, ↓. Upon further actions of the Hamiltonian, the emitted +− or −+

dipole in Eq. (5.44) can propagate in the spin background to the left or to the

right, leaving behind a free + fracton and resulting in one of the following two

configurations:

|ψ2〉 =

|∗ · · · ∗+− ↓ · · · ↓ + ∗ · · · ∗〉|∗ · · · ∗+ ↑ · · · ↑ −+ ∗ · · · ∗〉, (5.45)

where ∗ =↑, ↓ such that N↑ = N↓. With either a string of ↓’s or ↑’s (upper

and lower situation in Eq. (5.45) respectively), further actions of the Hamiltonian

enable the isolated fracton in Eq. (5.45) to move through the emission of an

additional dipole, which can then propagate in the spin background. This results

in configurations of the form:

|ψ3〉 =

|∗ · · · ∗+−+− ↓ · · · ↓ + ∗ · · · ∗〉

|∗ · · · ∗+ ↑ · · · ↑ −+−+ ∗ · · · ∗〉 ,(5.46)

where ∗ =↑, ↓ such that N↑ = N↓. Once configurations of the form Eq. (5.46) are

202

generated, a fracton can absorb a dipole when acted upon by the Hamiltonian, as

allowed by Eq. (5.12). The resulting configurations are of the form:

|ψ4〉 =

|∗ · · · ∗+ ↑ − ↓ · · · ↓ + ∗ · · · ∗〉|∗ · · · ∗+ ↑ · · · ↑ − ↓ + ∗ · · · ∗〉 ,(5.47)

where ∗ =↑, ↓ such that N↑ = N↓. Following the above discussion, one can show

that the repeated emission and absorption of multiple dipoles generates product

states within the Krylov subspace that are necessarily of the form:

|· · · ∗+ ↑ · · · ↑ − ↓ · · · ↓ + ↑ · · · ↑ − ↓ · · · ↓ + ∗ · · · 〉 , (5.48)

i.e. with strings of only ↑’s or ↓’s between consecutive fractons. Given the symme-

tries of the Hamiltonian, only strings of the form Eq. (5.48), that have the same

(Q,D, Sz) quantum numbers as the root state |ψ0〉, are allowed in the Krylov

subspace. Hence, this subspace is characterized by the presence of an emergent

string-order (equivalently, it is non-locally constrained).

To illustrate the novel features of this Krylov subspace, we compare the time

evolution of the charge density on the middle site (the site on which the fracton

resides initially) with that on a different site, which initially hosts a spin. The

results are shown in Fig. 5.2(a), which compares the charge density at the middle

site (in blue) to that at a different site (in green) as a function of time. Irrespective

of the spin configuration in the initial state, we consistently find that the middle

site exhibits a higher charge density as compared to any other site. Moreover,

as shown in Fig. 5.2(b), we find that this late-time charge density matches that

predicted by ETH, assuming the initial state lies in the middle of the spectrum of

203

the Krylov subspace. The charge density at an inverse temperature β restricted

to the Krylov subspace K is then given by

〈Qmid〉β =Tr(Qmide

−βH|K

)Tr(e−βH|K

) , (5.49)

where H|K is the restriction of the Hamiltonian H to the Krylov subspace K, and

Qmid is the charge operator of the middle site, using the established convention:

spins are charge neutral, whereas + and − fractons have charges +1 and −1

respectively.

Assuming infinite temperature (β = 0) in Eq. (5.49), we obtain [88]

〈Qmid〉β =Tr(Qmid

)Tr (1|K)

≡ QNDN

=3

N, (5.50)

where 1|K is the identity restricted to the Krylov subspace K, and thus Tr (1|K) =

DN , the Hilbert space dimension of K (H, |ψ0〉) of the chain of N sites. On the

other hand, the late time expectation value of the charge density on any other

site in the middle of the chain is 1/N [88]. We dub this phenomenon as quasi-

localization of the fracton, since it is localized for any finite system size although

the localization vanishes in the thermodynamic (N → ∞) limit. We empha-

size that unlike usual mechanisms for localization, which rely on the existence of

localized eigenstates [38, 86, 87], the phenomenon here is quasi-localization from

thermalization, which is a consequence of ergodicity, albeit ergodicity within a

constrained Krylov subspace.

204

5.8 Conclusions and Open Questions

In this chapter, we have studied a simple translation invariant model which con-

serves both charge and center-of-mass, and which provides a natural platform

for realising the physics of fractonic systems. Specifically, we find that the pair-

hopping model Eq. (5.1) exhibits the phenomenon of Krylov fracture, wherein

various regions of Hilbert space are dynamically disconnected even if they belong

to the same global symmetry sectors. In addition to exponentially many product

eigenstates, whose effect on quantum dynamics was studied in Refs. [86, 87], the

pair-hopping model also hosts several large closed Krylov subspaces with dimen-

sions that grow exponentially in the system size at half-filling.

We find that exponentially many of such large Krylov subspaces admit a map-

ping onto spin-1/2 XX models of various sizes and hence, constitute examples

of integrable Krylov subspaces. However, not all large Krylov subspaces show

signs of integrability; instead, the model also possesses exponentially many non-

integrable subspaces, many of which show level-repulsion and behaviour consis-

tent with ETH. Moreover, some of these Krylov subspaces are highly constrained,

which leads to atypical dynamical behaviour even within a thermal Krylov sub-

space, an effect we dub “quasilocalization due to thermalization”. By this, we

specifically mean that the late-time expectation values of local operators within

such subspaces deviate from the expected behaviour in generic translation invari-

ant systems. Finally, since the pair-hopping model appears as the leading order

hopping term in the strong-field limit of the interacting Wannier-Stark problem,

we make contact between our work and Bloch MBL. Besides shedding new light

on Bloch MBL, our work hence also provides an experimentally relevant setting

205

for studying the dynamics of center-of-mass preserving systems.

Our results, which illustrate the rich structure that can arise as a consequence of

Krylov fracture, harbour several implications for the dynamics of isolated quantum

systems. Firstly, in the presence of Krylov fracture, we have demonstrated that

notions of ergodicity and its violation are well-defined once restricted to large

Krylov subspaces. Moreover, we showed that usual diagnostics, such as energy

level-statistics, accurately capture whether such Krylov subspaces are integrable

or not. These results thus suggest that a modified version of ETH, restricted to

large Krylov subspaces, holds for systems with fractured Hilbert spaces.

Secondly, our results provide a clear example of a “semi-integrable” model i.e.,

one where integrable as well as non-integrable exponentially large Krylov sub-

spaces co-exist [85, 208]. When viewed from the perspective of the entire Hilbert

space (within a particular symmetry sector), the integrable Krylov subspaces are

examples of quantum many-body scars, since they are ETH-violating states em-

bedded within the entire many-body spectrum. Unlike the exponentially many

static configurations (one-dimensional Krylov subspaces) which necessarily exist

for any center-of-mass (dipole moment) conserving Hamiltonian [86, 87], these

integrable subspaces have an exponentially large dimension, which can lead to

non-trivial dynamics in an otherwise non-integrable model. For the cognoscenti,

we note that such subspaces are qualitatively distinct from subspaces generated

from states containing a blockaded region [231]. Thus, the existence of such inte-

grable Krylov subspaces of dimension much smaller than that of the full Hilbert

space, even if only approximately closed, might be related to quantum many-body

scars which by now have been observed in several constrained systems, including

the PXP model [66, 77]. Additionally, even large non-integrable subspaces show

206

ergodicity breaking with respect to the entire Hilbert space [86] and instead obey

ETH only once restricted to the Krylov subspace, resulting in highly non-general

thermal expectation values of local operators within such Krylov subspaces. Note

also that we have only focused on Krylov subspaces generated by root states that

are product states (see Eq. (5.19)), but one could also study closed Krylov sub-

spaces generated by other low-entanglement states; whether this leads to further

fracturing within the Krylov subspaces of the pair-hopping model is a question

for future work.

207

6Approximate Quantum Scars in a

Fractured Model

6.1 Introduction

Approximate quantum scars were observed as non-thermal oscillations after a

quench in cold atom experiments with Rydberg atoms [6], where the Hamiltonian

imposes a penalty on neighboring atoms can both be excited [183, 232], form-

208

ing an effective low-energy constrained Hilbert space. To explain the oscillations

from a Néel-like state, Refs. [66] and [68] studied the so-called PXP model, a toy

model for Rydberg atoms [233], and reported the presence of strong-ETH violat-

ing eigenstates that numerically are found to have a sub-thermal (logarithmic)

growth of entanglement entropy. The many-body analogue of classical phase

space was conjectured to be the Time-Dependent Variation Principle (TDVP)

manifold of states [234], a correspondence that was illustrated using Matrix Prod-

uct States (MPS) for the PXP model [70]. However, several questions about

these approximate quantum scars in constrained models remain open, including

their origin [69, 77, 82, 83, 86, 87, 184], the nature of the scars [72, 75, 76], and con-

structions of scars in fermionic systems. It is thus important to search for other

constrained systems that exhibit similar features. A natural system to look for

constrained dynamics is in models that exhibit Krylov fracture discussed in Chap-

ter 5. In this chapter, we explore interesting features restricted to some particular

non-integrable Krylov subspaces of these models, and show the appearance of

constrained Hilbert spaces and approximate quantum many-body scars.

This chapter is organized as follows. In Sec. 6.2, we show the emergence of

constrained Krylov subspaces at filling factors ν = p/(2p + 1). There we discuss

the mapping of a particular constrained subspace to the PXP model at filling

ν = 1/3, and the effective spin-chains models that arise at filling ν = p/(2p + 1)

in Sec. 6.2. We also discuss the properties of the constrained Hilbert spaces

that arise out of this models. In Sec. 6.3, we discuss the symmetries and non-

integrability of the constrained Hamiltonians such as symmetries and zero-modes.

In Sec. 6.5, we discuss the Forward Scattering Approximation for the models at

filling ν = p/(2p + 1) and show the existence of approximate many-body scars

209

that manifest in slow thermalizing product states. These initial states are charge

density waves in the quantum Hall language. In Sec. 6.5.1, we briefly discuss

the stability of scars to electrostatic terms that arise in the quantum Hall setup.

There we numerically present some evidence that these many-body scars survive

for small strengths of electrostatic terms.

6.2 Effective Spin-Chains and Constrained Hilbert spaces

We now review the mapping of the Hamiltonian of Eq. (5.1) on to spin-1 chain

models, first discussed in Ref. [225]. In this work, we restrict ourselves to filling

factors of the form ν = p/(2p + 1) and system sizes of the form L = (2p + 1)N ,

N ∈ N. We numerically observe that the ground state of H is at half filling for

even L, and filling ν = (L± 1)/(2L) for odd L. Thus the sectors we study are in

the middle of the full spectrum of H.

6.2.1 Mapping onto spin-1 chains

Here we provide a summary of the mapping, which we illustrate with the details

in Secs. 6.2.2 and 6.2.3. For the pair-hopping model at filling ν = p/(2p+ 1), we

focus on the Krylov subspace K(p), defined as

K(p) ≡ K(∣∣R(p)

⟩, H),∣∣R(p)

⟩=

N⊗j=1

∣∣∣ 0(10)p⟩, (6.1)

where (01)p denotes the repetition of (01) p times. Each of the N boxed units

in Eq. (6.1) is referred to as a unit cell (the system size L = (2p + 1)N). For

example, for p = 1 (ν = 1/3) and p = 2 (ν = 2/5), the states∣∣R(1)

⟩and

∣∣R(2)⟩

210

read

∣∣R(1)⟩=∣∣∣ 010 · · · 010

⟩,∣∣R(2)

⟩=∣∣∣ 01010 · · · 01010

⟩. (6.2)

To study the dynamics of states in K(p) under the Hamiltonian H, it is sufficient

to study the eigenstates of H(p), the restriction of H to K(p).

Since the Hamiltonian H is a four-site Hamiltonian, in terms of unit cells, the

H(p) is a two unit cell Hamiltonian that has both intra- and inter- unit cell terms.

That is, it is of the form

H(p) =N∑j=1

H(p)j +

Nb∑j=1

H(p)j,j+1, (6.3)

where H(p)j and H(p)

j,j+1 are one unit cell and two unit cell terms respectively, and

Nb = N − 1 and Nb = N for OBC and PBC respectively. We will obtain the

explicit expressions for H(p)j and H(p)

j,j+1 for various p in Secs. 6.2.2 and 6.2.3. As

we will show there, H(p) can be mapped onto a spin-1 Hamiltonian with each unit

cell (as defined in∣∣R(p)

⟩in Eq. (6.1)) replaced by p spin-1’s. ∗ This mapping

proceeds by appropriately inserting (p− 1) fictitious 0’s (pseudozeroes) into each

unit cell, grouping the resulting 3p sites into p blocks of 3, and identifying each

block with one of the following spin-1 configurations [225]

|o〉 ≡ |0 1 0〉 |+〉 ≡ |0 0 1〉 |−〉 ≡ |1 0 0〉 . (6.4)

As we will see in Secs. 6.2.2 and 6.2.3, these three spin-1 configurations are suffi-∗For PBC, the (2p+1) different ways of grouping the sites into unit cells results in an effective

Hamiltonian with the same spectrum since H is a center-of-mass preserving Hamiltonian thathas a (2p+ 1)-fold degenerate spectrum. [214]

211

cient to obtain a faithful mapping.

In the rest of the chapter, we denote configurations of N unit cells as |σ〉 =

|σ1σ2 · · ·σN〉, where σj is the configuration of spin-1’s in the j-th unit cell (i.e.

σj encodes the configuration of p spin-1’s). For example, when N = 2, p = 3

(ν = 3/7), consider the configuration |ψ〉 =∣∣∣ 0011010 0110010

⟩, consisting of

14 orbitals. As we will explain in Sec. 6.2.3, this configuration can be uniquely

mapped onto the spin-1 configuration |σ〉 =∣∣ +oo o−o

⟩, which consists of

N = 2 unit cells and p = 3 spin-1’s in each unit cell. Then, we denote the

configuration of each unit cell by the three spin configuration σ1 = +oo and

σ2 = o−o . We denote the spin-1 operators acting on the n-th spin on the j-th

unit cell as Sαj,n, where α = x, y, z,+,−. Further, we define operators Tj,n and

Uj,n that will be used frequently throughout our analysis:

Tj,n ≡Szj,nS

−j,n√2

, Uj,n ≡S−j,nS

zj,n√2

. (6.5)

The only non-vanishing actions of the operators Tj,n and Uj,n are given by Tj,n |o〉 =

− |−〉 and Uj,n |+〉 = |o〉, where |∗〉j,n, with ∗ = +, o,− is the configuration of the

spin-1 on the n-th site in the j-th unit cell.

Further, given a configuration |σ〉 = |σ1σ2 · · · σN〉, we define the following quan-

tities that we use later in the chapter.

Pσj =

p∑l=1

δσjl,+, Mσj =

p∑l=1

δσjl,−, X(P )σj

=

p∑l=1

(p+ 1− l) δσjl,+, X(M)σj

=

p∑l=1

l δσjl,−,

(6.6)

where σjl is the configuration of the l-th spin in the j-th unit cell. Pσj (resp. Mσj)

is the number of +’s (resp. −’s) in the j-th unit cell, X(P )σj (resp. X(M)

σj ) is the sum

212

of positions of +’s (resp. −’s) within the j-th unit cell counted from right (resp.

left). For example, if σj =1 2 3 4 5 6 7

o−−+o++ , we have Pσj = 3, Mσj = 2, X(P )σj = 7,

and X(M)σj = 5.

6.2.2 Filling ν = 1/3

Effective Hamiltonian

We now derive the effective Hamiltonian H(1) at filling ν = 1/3, i.e. p = 1, that

acts on the Krylov subspace K(1). Consider the pair-hopping Hamiltonian on a

system of size L = 3N . Using the spin-1 mapping of Eq. (6.4), the state∣∣R(1)

⟩of

Eq. (6.2) in the spin-1 language with the same choice of unit cells reads

∣∣R(1)⟩= | o o · · · o 〉 . (6.7)

Using Eqs. (5.1) and (5.2), acting on∣∣R(1)

⟩with H once results in a sum of config-

urations that have one pair of | + − 〉 on neighboring unit cells in a “vacuum”

of unit cells in the configuration | o 〉, i.e. configurations of the form

| o o · · · o + − o · · · o 〉 . (6.8)

We immediately deduce that H(1)j = 0 and that the actions of a single term H(1)

j,j+1

on the spin-1 configurations read

|· · · o + · · · 〉H(1)

j,j+1−−−−→ 0,j j+1

|· · · − + · · · 〉H(1)

j,j+1−−−−→ 0,j j+1

|· · · − o · · · 〉H(1)

j,j+1−−−−→ 0j j+1

|· · · o o · · · 〉H(1)

j,j+1←−−→j j+1

|· · · + − · · · 〉. (6.9)

213

Using Eq. (6.9), further actions of H(1) on the state of Eq. (6.8) either (i) destroy

the nearest neighbor configuration + − , or (ii) create the nearest neighbor state

+ − from the configuration o o . Thus, one never obtains the nearest neighbor

configurations + + , + o , o − , or − − , and it is sufficient to consider

the rules of Eq. (6.9). Note that (i) and (ii) are the only possible actions of H(1)

on subsequent configurations as well. In the language of the original orbitals,

these processes correspond to squeezing and antisqueezing of close configurations

respectively. Thus any state |σ〉 = |σ1 · · ·σN〉 in the Krylov subspace K(1) obeys

the following constraints:

(c1) The only allowed configuration of nearest neighbor unit cells are

+ − , o o , o + , − o , − + . (6.10)

This constraint can be compactly stated as Pσj =Mσj+1∀j, 1 ≤ j ≤ N − 1.

(c2) With OBC, within the Krylov subspace, the leftmost (resp. rightmost) unit

cell cannot have the configuration − (resp. + ), i.e. in the Krylov subspace

Mσ1 = 0 (resp. PσN = 0). This follows from Eq. (6.9) where + and − are

created together only on nearest neighboring unit cells with + in the left

unit cell and − in the right unit cell.

This is an example of a constrained Hilbert space. Thus, using Eq. (6.9), H(1)

reads [225]

H(1) = −Nb∑j=1

(U †j,1Tj+1,1 + h.c.

). (6.11)

Note that although the subscript “· · · , 1” in the spin operators is redundant for

this case because the unit cell contains a single spin-1, we continue to use it in

214

order to smoothly transition to arbitrary values of p.

Mapping on to the PXP model

The constrained Hilbert space K(1) can be alternately specified by moving to the

dual lattice of the spin-1 lattice, i.e. the sites j + 12 defined on the bonds

(j, j + 1). Thanks to the highly constrained Hilbert space, configurations of N

unit cells in K(1) can be written in terms of spin-1/2 degrees of freedom on the

dual lattice of N − 1 (resp. N) sites for OBC (resp. PBC) using the mapping

| + − 〉j,j+1 → | ↑ 〉j+ 12| o + 〉j,j+1 → | ↓ 〉j+ 1

2| − o 〉j,j+1 → | ↓ 〉j+ 1

2

| o o 〉j,j+1 → | ↓ 〉j+ 12, | − + 〉j,j+1 → | ↓ 〉j+ 1

2, (6.12)

where | ∗ ∗ 〉j,j+1 is the configuration of the j-th and (j+1)-th unit cells on the

spin-1 lattice and |∗〉j+ 12

is the configuration of the site j + 12

on the dual lattice.

The subscripts are taken to be modulo N for PBC. In other words, the nearest

neighbor configuration + − maps onto ↑ whereas all other nearest neighbor

configurations in the Krylov subspace map onto ↓. While the mapping appears

to be many to one, we will shortly show that it is in fact invertible for both PBC

and OBC as a result of the constraints of K(1). For example, the configuration

|ψ1〉 =1 2 3 4 5 6

| o + − + − o 〉 (6.13)

215

maps on to the configurations∣∣ψ(PBC)

⟩and

∣∣ψ(OBC)⟩

for PBC and OBC respec-

tively, where

∣∣ψ(PBC)⟩=

32

52

72

92

112

132

| ↓ ↑ ↓ ↑ ↓ ↓ 〉,∣∣ψ(OBC)

⟩=

32

52

72

92

112

| ↓ ↑ ↓ ↑ ↓ 〉. (6.14)

Note that the mapping of Eq. (6.12) does not allow the dual lattice configuration

| ↑ ↑ 〉 even though it includes all possible nearest neighbor configurations allowed

in K(1) (Eq. (6.10)). Thus, the constraint (c1) on K(1) defined in Sec. 6.2.2 trans-

lates to the constraint that no nearest neighbor spins can be ↑ in the dual lattice

(a hallmark of the PXP model [66]). The mapping from the dual lattice back to

the spin-1 lattice reads

| ↓ ↑ 〉j− 12,j+ 1

2→ | + 〉j , | ↑ ↓ 〉j− 1

2,j+ 1

2→ | − 〉j , | ↓ ↓ 〉j− 1

2,j+ 1

2→ | o 〉j ,

(6.15)

where | ∗ ∗ 〉j− 12,j+ 1

2, is the configuration of the (j − 1/2)-th and (j + 1/2)-th

sites on the dual lattice and |∗〉j, ∗ = +, o,− is the configuration of the j-th unit

cell on the spin-1 lattice. The subscripts are taken to be modulo N for PBC. Note

that Eqs. (6.12) and (6.15) ensure that the mapping is one to one for PBC. For

example, the configuration∣∣ψ(PBC)

⟩of Eq. (6.14) maps onto |ψ1〉 of Eq. (6.13)

under Eq. (6.15).†

With OBC, Eq. (6.15) can be applied to obtain the configuration of the unit

cells j, 2 ≤ j ≤ N − 2. The configurations of the leftmost (j = 1) and rightmost

(j = N) unit cells can then be uniquely obtained using the constraint (c2) defined†Note that since the leftmost and rightmost sites in the spin-1 language are labelled by j = 1

and j = N respectively, the leftmost and rightmost sites on the dual lattice are labelled by j = 32

and j = N + 12 for PBC (j = N − 1

2 for OBC).

216

in Sec. 6.2.2. For example, using the rules of Eq. (6.15), the configuration∣∣ψ(OBC)

⟩of Eq. (6.14) maps onto | ∗ + − + − ∗ 〉, and the ∗ on the leftmost and

rightmost unit cells are o , since that is the only allowed configuration allowed

by the constraints (c1) and (c2). Thus∣∣ψOBC⟩ of Eq. (6.14) maps onto |ψ1〉 of

Eq. (6.13).

Since the Hamiltonian H(1) consists of two unit cell terms, using the mapping

of Eq. (6.12), the corresponding Hamiltonian H(d) in the dual lattice consists of

three site terms H(d)

j− 12,j+ 1

2,j+ 3

2

in the bulk and two site terms on the boundaries

H(d)

j− 12,j+ 1

2

. For example, the non-vanishing actions of H(1)j,j+1 in the bulk of the

chain translate to

H(1)j,j+1

j j+1

| ∗ + − ∗ 〉 =j j+1

| ∗ o o ∗ 〉 =⇒ H(d)

j− 12,j+ 1

2,j+ 3

2

j+ 12

| ↓ ↑ ↓ 〉 =j+ 1

2

| ↓ ↓ ↓ 〉,

H(1)j,j+1

j j+1

| ∗ o o ∗ 〉 =j j+1

| ∗ + − ∗ 〉 =⇒ H(d)

j− 12,j+ 1

2,j+ 3

2

j+ 12

| ↓ ↓ ↓ 〉 =j+ 1

2

| ↓ ↑ ↓ 〉,(6.16)

where ∗ corresponds to any allowed configuration of the unit cell, and subscripts

are taken modulo N for PBC. However with OBC, the actions of H(1)j,j+1 on the

left and right boundaries read (using the constraint (c2))

H(1)1,2

1 2

| + − ∗ 〉 =1 2

| o o ∗ 〉 =⇒ H(d)32, 52

32

52

| ↑ ↓ 〉 =32

52

| ↓ ↓ 〉,

H(1)1,2

1 2

| o o ∗ 〉 =1 2

| + − ∗ 〉 =⇒ H(d)32, 52

32

52

| ↓ ↓ 〉 =32

52

| ↑ ↓ 〉,

H(1)N−1,N

N−1 N

| ∗ + − 〉 =N−1 N

| ∗ o o 〉 =⇒ H(d)

N− 32,N− 1

2

N− 32N− 1

2

| ↓ ↑ 〉 =N− 3

2N− 1

2

| ↓ ↓ 〉 ,

H(1)N−1,N

N−1 N

| ∗ o o 〉 =N−1 N

| ∗ + − 〉 =⇒ H(d)

N− 32,N− 1

2

N− 32N− 1

2

| ↓ ↓ 〉 =N− 3

2N− 1

2

| ↓ ↑ 〉 ,

(6.17)

217

The terms of the dual lattice Hamiltonian in Eqs. (6.16) and (6.17) are terms the

PXP model, studied in several contexts in the literature [66, 71, 235] i.e.

H(d) =

N+ 1

2∑l= 3

2

Pl−1σxl Pl+1 if PBC

N− 32∑

l= 52

Pl−1σxl Pl+1 + σx3

2

P 52+ PN− 3

2σxN− 1

2

if OBC(6.18)

where σxl is a Pauli matrix on site l, and Pl is a projector on site l on to |↓〉,

i.e. Pl ≡ (1−σzl )

2. Thus, the pair-hopping Hamiltonian H restricted to the Krylov

subspace K(1) is exactly the PXP Hamiltonian. In Sec. 6.5 we will rely on this

mapping and show the existence of quantum many-body scars [66,68] in the pair-

hopping Hamiltonian H. To easily generalize to other filling factors, even when

p = 1 we stick to the original spin-1 degrees of freedom language instead of the

spin-1/2 degrees of freedom of the PXP model. The dimension of the Krylov

subspace K(1) can be shown to be D(1)N = FN+1, where Fn is the n-th Fibonacci

number. Thus K(1) is isomorphic to the Hilbert space of the PXP model, [66]

and D(1)N thus scales as φN , where the quantum dimension φ is the Golden ratio

φ = 1+√5

2.

6.2.3 Filling ν = p/(2p+ 1)

We now move to the effective Hamiltonians at filling factors ν = p/(2p + 1)

of the original Hamiltonian H. Here we focus on the Krylov subspace K(p) =

K(∣∣R(p)

⟩, H)

defined in Eq. (6.1). To understand the structure of K(p), we invoke

an important property shown in Ref. [225] (see Eq. (15) and Appendix B therein).

218

First note that∣∣R(p)

⟩of Eq. (6.1) is of the form

∣∣R(p)⟩=

∣∣∣∣· · · i0(10)q · · ·⟩ (6.19)

for some i, q, where i, 1 ≤ i ≤ L for PBC (or 1 ≤ i ≤ L − 2 for OBC) denotes

the position of the orbital, and q, 1 ≤ q ≤ p denotes the number of times the

pattern “10” appears consecutively. As a consequence of the squeezing property

of Eq. (5.2), this pattern implies that [225]

i+2q∑k=i

〈ψ| c†kck |ψ〉 ≥ q for any |ψ〉 ∈ K(p). (6.20)

This property constrains the allowed unit cell configurations in K(p). For example,

for ν = 2/5 (p = 2),∣∣R(2)

⟩is defined in Eq. (6.2), and all the unit cells configu-

rations read 01010 . According to Eq. (6.20), any unit cell for some |ψ〉 ∈ K(2) of

the form n1n2n3n4n5 should satisfy n1 + n2 + n3 ≥ 1 (resp. n3 + n4 + n5 ≥ 1),

which is obtained by choosing q = 1 and i in Eq. (6.20) to be the first (resp.

third) site within any unit cell of∣∣R(2)

⟩of Eq. (6.1). From these inequalities, we

deduce that the unit cell configurations 00011 (resp. 11000 ) violate Eq. (6.20),

and are thus not allowed for any configuration in K(2). To summarize, we obtain

eight allowed unit cell configurations for p = 2:

00110 01100 10100 00101 01010 10010 01001 10001 . (6.21)

The unit cell configurations of Eq. (6.21) can be uniquely mapped onto config-

urations of two spin-1’s by adding one fictitious pseudozero in between any two

219

consecutive (although not necessarily adjacent) 1’s in Eq. (6.21) [225]:

00110 = 001[0]10 ≡ +o 01100 = 01[0]100 ≡ o−

10100 = 10[0]100 ≡ −− 00101 = 001[0]01 ≡ ++

01010 = 01[0]010 ≡ oo 10001 = 100[0]01 ≡ −+

01001 = 010[0]01 ≡ o+ 10010 = 100[0]10 ≡ −o

(6.22)

where +, − and o are spin-1 configurations defined in Eq. (6.4) and [0] is the

pseudozero. The addition of pseudozeroes and mapping on to spin-1’s can be

reversed by deleting a 0 between the 1’s within a unit cell after inverting the

spin-1 mapping using Eq. (6.4). For example, the configuration o+ maps onto

010001 using Eq. (6.4), which corresponds to 01001 since we know one of the 0’s

between the 1’s is a pseudozero. Similarly, for general p, z = p − 1 pseudozeroes

are added between two consecutive 1’s so that the size of the configuration in

each unit cell is 3p, which can then be mapped on to a configuration of p spin-1’s

using Eq. (6.4). Such a mapping is one to one as a consequence of Eqs. (6.19)

and (6.20), and we refer readers to Ref. [225] for a complete discussion of this

property.

The action of the Hamiltonian H in Eq. (5.1) can be written in terms of spin-1

variables using the mapping of Eq. (6.4). Here we show this action separately

when the term Hj in Eq. (5.1) acts between neighboring unit cells, and when it

acts within a unit cell. When Hj acts between neighboring unit cells,

j j+3∣∣∣ · · · 1 0 0 1 · · ·⟩

Hj←→j j+3∣∣∣ · · · 0 1 1 0 · · ·

⟩. (6.23)

Using the spin-1 mapping of Eq. (6.4), and noting that configurations within

220

a unit cell do not have adjacent 1’s (“11”) after the addition of pseudozeroes

(hence · · · 10 and 01 · · · in Eq. (6.23) respectively read · · · 010 and 010 · · ·

or · · · [0]10 and 01[0] · · · after the addition of pseudozeroes), the action of the

effective Hamiltonian reads

j j+1

| · · · o o · · · 〉H(p)

j,j+1←−−→j j+1∣∣ · · ·+ − · · ·

⟩, (6.24)

where j is the unit cell index. Similarly, when Hj acts within a unit cell,

j j+3∣∣∣ · · · 0 1 [0] 1 0 · · ·⟩

Hj←→j j+3∣∣∣ · · · 1 0 [0] 0 1 · · ·

⟩. (6.25)

Using Eqs. (6.25) and (6.4), relying once again on the absence of adjacent 1’s

(“11”) after the addition of pseudozeroes, the action in the spin-1 language thus

reads

n n+1∣∣ · · · o + · · ·⟩ (

H(p)j

)n,n+1←−−−−−−→

n n+1∣∣ · · ·+ o · · ·⟩,

n n+1∣∣ · · · o− · · · ⟩ (H(p)

j

)n,n+1←−−−−−−→

n n+1∣∣ · · · − o · · ·⟩, (6.26)

where j is the unit cell index. The effective Hamiltonian within K(p) can thus be

written as [225]

H(p) =N∑j=1

[p−1∑n=1

(T †j,nTj,n+1 + U †

j,nUj,n+1 + h.c.)−(U †j,pTj+1,1 + h.c.

)](6.27)

where Tj,n and Uj,n are spin-1 operators defined in Eq. (6.5). Note that Eq. (6.27)

reduces to Eq. (6.11) when p = 1.

221

We now describe the Krylov subspace K(p). After adding (p− 1) pseudozeroes

in between two 1’s within a unit cell (and not between 1’s in different unit cells),∣∣R(p)⟩

of Eq. (6.1) reads

∣∣R(p)⟩=

N⊗j=1

∣∣∣ (01[0])p−1010⟩, (6.28)

where (01[0])p−1 indicates that (01[0]) is repeated p − 1 times. Thus, using

Eq. (6.4), in terms of spin-1 variables∣∣R(p)

⟩of Eq. (6.28) reads

∣∣R(p)⟩= | o · · · o · · · o · · · o 〉 . (6.29)

As a consequence of the Eqs. (6.24), (6.26) and (6.29), any configuration of N

unit cells |σ〉 = |σ1σ2 · · ·σN〉 in K(p) has the following constraints:

(c1) Pσj = Mσj+1, where Pσj (resp. Mσj+1

) is the number of + (resp. −) in the

j-th (resp. (j+1)-th unit cell): This follows from Eq. (6.24), where starting

from∣∣R(p)

⟩of Eq. (6.29), + and − are created together on neighboring unit

cells.

(c2) Within each unit cell − appears to the left of +: This follows from Eq. (6.24)

because starting from∣∣R(p)

⟩, + and − are always created to the left and

right of the unit cells respectively, and they cannot cross each other due to

Eq. (6.26). That is, there is no term of the Hamiltonian that allows the

process +− ↔ −+ within a unit cell.

(c3) With OBC, the leftmost (resp. rightmost) unit cell σ1 (resp. σN) cannot

have a − (resp. +), i.e. Mσ1 = 0 (resp. PσN = 0): This follows from

222

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

∆E

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

P(∆E

)15 10 5 0 5 10 15

E

0.00

0.05

0.10

0.15

0.20

0.25

ρ(E

)

Figure 6.1: Level statistics of the eigenstates of the Hamiltonian H(2). The blackline shows the expected GOE curve for non-integrable models. The standardparameter 〈r〉 ≈ 0.5284, close to the GOE value of 〈r〉 ≈ 0.5295. (Inset) Densityof states for the eigenstates of the Hamiltonian H(2). The peak at E = 0 indicatesthe presence of a large number of zero-modes in an otherwise non-integrable modelwith a Gaussian density of states. Data is shown for a system with p = 2 andN = 12 in the quantum number sector (k,X) = (0,+1), where X is the quantumnumber corresponding to the symmetry IP .

Eq. (6.24), where starting + and − are created together on neighboring

unit cells with + in the left unit cell and − in the right unit cell, and thus

the leftmost (resp. rightmost) unit cell cannot have a − (resp. +).

The expression for the Hilbert space dimension D(p) of K(p) can be obtained for

OBC. For general p and large N , D(p) grows with N as D(p) ∼ (λ(p))N where the

quantum dimension λ(p) is given by

λ(p) ∼ 2p−1φ, (6.30)

where φ is the Golden ratio. We now review some important properties of the

Hamiltonians H(p) of Eq. (6.27).

223

Figure 6.2: The graph G(2) for N = 2 with PBC. Each node (ellipsoid) correspondsto a many-body basis state of the Krylov subspace K(2). A link is drawn betweentwo nodes if there exists a non-zero matrix element in the Hamiltonian that couplesthe corresponding many-body states. Nearest neighbor nodes have charges Q(2)

that differ by 1. The red and blue nodes represent the ones with C(2)σ = +1 andC(2)σ = −1 respectively.

6.3 Symmetries and Non-integrability of the Effective

Hamiltonians

We first discuss the symmetries of the Hamiltonian H(p). Consider the transfor-

mation of the Hamiltonian H(p) under spin flips of spin-1’s in the chain, given by

the unitary operator

P =N∏j=1

p∏n=1

exp(iπSxj,n

). (6.31)

Since PS±j,nP† = S∓

j,n and PSzj,nP† = −Szj,n, the operators Tj,n and Uj,n of Eq. (6.5)

transform as

PTj,nP† = −U †j,n, PUj,nP† = −T †

j,n. (6.32)

Thus, under spin-flips the Hamiltonian H(p) transforms to

PH(p)P† =N∑j=1

[p−1∑n=1

(Uj,nU

†j,n+1 + Tj,nT

†j,n+1 + h.c.

)−(Tj,pU

†j+1,1 + h.c.

)](6.33)

Further, we consider inversion symmetry I, which transforms operators Tj,n and

Uj,n as

ITj,nI† = TN+1−j,p+1−n, IUj,nI† = UN+1−j,p+1−n. (6.34)

224

Acting I and I† on the left and right of Eq. (6.33), after rearranging the sum we

obtain

IPH(p)P†I† =N∑j=1

[p−1∑n=1

(T †j,nTj,n+1 + U †

j,nUj,n+1 + h.c.)−(U †j,pTj+1,1 + h.c.

)]= H(p)

(6.35)

Thus, the Hamiltonian H(p) has a Z2 symmetry generated by the unitary IP .

We denote its quantum numbers by X = ±1. The IP symmetry of the Hamil-

tonians H(p) is the same as inversion symmetry of the pair-hopping Hamiltonian

H. In the original language, this symmetry is the inversion symmetry of the pair

hopping Hamiltonian H of Eq. (5.1). We illustrate this with a simple example

with N = 2 and p = 2. Under the operator IP , the configuration∣∣ o+ o−

⟩transforms to

∣∣ +o −o⟩. Using the mapping of Eq. (6.22), in the original lan-

guage these states read∣∣∣ 010[0]01 01[0]100

⟩(resp.

∣∣∣ 01001 01100⟩

) and∣∣∣ 001[0]10 10[0]010⟩

(resp.∣∣∣ 00110 10010

⟩) with (resp. without) the pseu-

dozeroes, which are related by the action of inversion.

With PBC, H(p) is also trivially invariant under a translation by one unit

cell. Thus it can be block-diagonalized into N blocks labelled by momenta

k = 2πj/N, 0 ≤ j ≤ N − 1. However, the translation operator T and I

(hence IP) do not commute unless k = 0 or k = π.

The Hamiltonians H(p) are non-integrable. The characteristic property of non-

integrable models is the appearance of Wigner-Dyson energy level statistics within

a given quantum number sector. When p = 1, H(1) can be exactly mapped on to

the PXP model as discussed in Sec. 6.2.2, and the Wigner-Dyson level statistics

of the PXP model was observed in Ref. [66]. We find similar level statistics for all

the quantum number sectors of the Hamiltonian H(p) for p ≤ 3 up to the system

225

sizes we are able to study numerically. In Fig. 6.1, we show the level statistics

in the (k,X) = (0,+1) sector of the Hamiltonian H(2) for a system with N = 12

unit cells, where X is the quantum number corresponding to the IP symmetry.

6.4 Quantum Many-Body Scars

6.4.1 Charge Operators

To unravel the properties relevant for many-body scars, we map H(p) onto a

single particle hopping on a graph G(p), where each node of the graph represents a

product configuration |σ〉 ∈ K(p). This idea was employed to study the PXP model

in Refs. [66] and [68]. The links of the graph indicate the non-vanishing matrix

elements of the Hamiltonian between the corresponding node configurations. For

example, the graph G(2) for N = 2 with PBC is shown in Fig. 6.2. To better

understand the structure of the graph G(p), it is useful to define a charge Q(p)σ

associated with each node (each configuration |σ〉. We start with p = 1 as an

example. Here, we define a charge Q(1)σ as

Q(1)σ ≡

Nb∑j=1

(−1)j(Pσj +Mσj+1

2

). (6.36)

Note that Pσj = Mσj+1according to the constraint (c1) in Sec. 6.2.2. From

Eq. (6.9), we deduce that the action of each term in the Hamiltonian changes the

number of + spins in the j-th unit cell and the number of − spins in the (j+1)-th

unit cell by 1. That is, for product states |σ〉 and |τ〉, the following holds for a

226

single value of j∗, 1 ≤ j∗ ≤ Nb (since p = 1, Pσj ,Mσj ∈ 0, 1 ∀ σj):

H(1)j∗,j∗+1 |σ〉 = |τ〉 =⇒ Pτj∗ = 1− Pσj∗ ,Mτj∗+1

= 1−Mσj∗+1. (6.37)

Using Eqs. (6.36) and (6.37), the Hamiltonian H(1) can be split into two parts as

H(1) = H(1)+ +H(1)

− , (6.38)

where all the basis states |τ〉 that appear in H(1)+ |σ〉 (resp. H(1)

− |σ〉) satisfy Q(1)τ =

Q(1)σ + 1 (resp. Q(1)

τ = Q(1)σ − 1). Thus, H(1) can be written as a sum of charge-

raising and charge-lowering operators H(1)+ and H(1)

− respectively. In the operator

language, H(1)+ and H(1)

− read

H(1)+ = −

∑j even

U †j,1Tj+1,1−

∑j odd

T †j+1,1Uj,1, H

(1)− = −

∑j even

T †j+1,1Uj,1−

∑j odd

U †j,1Tj+1,1.

(6.39)

This splitting will be useful when we discuss quantum many-body scars in Sec. 6.5.

All of the structure of H(1) described in the previous paragraph generalizes to

any p. We now define charges Q(p)σ as

Q(p)σ ≡

Nb∑j=1

(−1)j+1

(Pσj +Mσj+1

2−(X(P )σj

+X(M)σj+1

)), (6.40)

where Pσj , Mσj , X(P )σj , and X

(M)σj are defined in Eq. (6.6). Note that Pσj =Mσj+1

according to the constraint (c1) in Sec. 6.2.3. For example, when p = 3 and N = 2

consider the configuration∣∣ o ++ −o−

⟩with PBC. Here, σ1 = o ++ and

σ2 = −o− . Using Eq. (6.6), we obtain Pσ1 = 2, Mσ1 = 0, X(P )σ1 = 3, X(M)

σ1 = 0,

227

and Pσ2 = 0, Mσ2 = 2, X(P )σ2 = 0, X(M)

σ2 = 4. Thus the charge of this configuration

is Q(p)σ = 5. Note that when p = 1, X(P )

σj = Pσj and X(M)σj = Mσj , and thus

Eq. (6.40) reduces to Eq. (6.36). When the charge is defined as in Eq. (6.40), we

can in fact write the Hamiltonian H(p) as

H(p) = H(p)+ +H(p)

− , H(p)± |σ〉 = |τ〉+ · · · =⇒ Q(p)

τ = Q(p)σ ± 1 (6.41)

for product configurations |σ〉 and |τ〉. The operator expressions for H(p)+ and H(p)

−

read [78]

H(p)+ = −

∑j even

U †j,pTj+1,1 −

∑j odd

T †j+1,1Uj,p +

∑j odd

p−1∑n=1

(U †j,n+1Uj,n + T †

j,n+1Tj,n

)+∑

j even

p−1∑n=1

(U †j,nUj,n+1 + T †

j,nTj,n+1

),

H(p)− = −

∑j even

T †j+1,1Uj,p −

∑j odd

U †j,pTj+1,1 +

∑j odd

p−1∑n=1

(U †j,nUj,n+1 + T †

j,nTj,n+1

)+∑

j even

p−1∑n=1

(U †j,n+1Uj,n + T †

j,n+1Tj,n

). (6.42)

6.5 Forward Scattering Approximation

We now discuss the fate of quantum many-body scars in the Hamiltonians H(p).

We first discuss the case of H(1), which maps on to the PXP model. In the PXP

model, the anomalous dynamics of the Néel state was studied, [66,68] which reads

(for PBC and even system size)

∣∣∣Z(PXP)2

⟩= | ↑ ↓ ↑ ↓ · · · ↑ ↓ 〉 . (6.43)

228

0 20 40 60 80 100T

0

2

4

6

8

10S

(a)

| (2)2

⟩|R(2)

⟩0 20 40 60 80 100

0

2

4

6

8

0 20 40 60 80 100T

0.0

0.2

0.4

0.6

0.8

1.0

|⟨ ψ|e−iH

(2)T|ψ⟩ |2

(b) | (2)2

⟩|R(2)

⟩

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

Figure 6.3: Time evolution of (a) entanglement entropies, and (b) fidelities ofthe

∣∣∣Z(2)2

⟩and

∣∣R(2)⟩

states (with k = 0) under the Hamiltonian H(2). Data isshown for PBC in the quantum number sector (k,X) = (0,+1) for N = 16 (insetsdisplay the same quantities for N = 14).

In particular, the entanglement growth of the∣∣ZPXP

2

⟩state for the PXP model

shows oscillations about a sub-thermal value in spite of the Wigner-Dyson level

statistics and thus the non-integrability of the PXP model. [66] This anomalous

behavior was explained by the existence of eigenstates in the PXP Hamiltonian

that have a subthermal entanglement entropy and a anomalously large overlap

with the∣∣∣Z(PXP)

2

⟩state for any finite system size. Such states were then approx-

imated using the so-called Forward Scattering Approximation (FSA), [66] which

we elaborate below for the Hamiltonians H(p).

Before we move on to general p, we translate the scar physics of the PXP

model in terms of the Hamiltonian H(1) and the Krylov subspace K(1), allowing

a direct generalization to arbitrary p. Using the mapping of Eq. (6.15), we map

the∣∣∣Z(PXP)

2

⟩state of Eq. (6.43) on to the

∣∣∣Z(1)2

⟩state of the constrained Hilbert

space K(1) defined in Eq. (6.1), which are density-wave configurations that reads

229

(for PBC and even N)

∣∣∣Z(1)2

⟩= | + − + · · · − + − 〉 =

∣∣∣ 001 100 001 · · · 100 001 100⟩,

(6.44)

which is the “maximally squeezed state” [236, 237] at ν = 1/3 in the quantum

Hall language, i.e. the configuration that cannot be “squeezed” further but it

can be “antisqueezed” (see Eq. (5.2)). (Note that in the quantum Hall case, the

presence of longer range squeezing terms leads to a different maximally squeezed

configuration). We can show that the∣∣∣Z(1)

2

⟩state of Eq. (6.44) is the state in

K(1), with the lowest charge Q(1)Z2

= −N/2. Since the PXP Hamiltonian maps

on to H(1), the anomalous dynamics of the∣∣∣Z(PXP)

2

⟩state of Eq. (6.43) for the

PXP model thus maps on to the dynamics of the∣∣∣Z(1)

2

⟩state of Eq. (6.44) for the

Hamiltonian H(1) in Eq. (6.11). Thus, generalizing to p > 1, we conjecture that

the Hamiltonian H(p) shows anomalous dynamics for the lowest charge states in

K(p), which are the∣∣∣Z(p)

2

⟩states that read

∣∣∣Z(p)2

⟩=

∣∣ + · · ·+ − · · ·− · · · + · · ·+ − · · ·−⟩

=∣∣∣ 00101 · · · 01 10 · · · 10100 · · · 00101 · · · 01 10 · · · 10100

⟩.(6.45)

We now demonstrate the FSA for H(p). Note that we directly work with general

p, and this analysis reduces to that of the PXP model in Refs. [66, 68] by setting

p = 1 and using the mapping of Eq. (6.12). We first construct the Krylov subspace

K(p)+ defined as

K(p)+ ≡ K

(∣∣∣Z(p)2

⟩,H(p)

+

), (6.46)

where H(p)+ is the charge-raising part of the Hamiltonian, shown in Eq. (6.41) and

230

Figure 6.4: (a) Overlap of the∣∣∣Z(2)

2 (k = 0)⟩

state with the eigenstates |ψ(E)〉 ofthe Hamiltonian. The vertical lines represent the energies of the scars as predictedby the FSA (i.e. eigenvalues of H(2)

FSA, see Eq. (6.50)) (b) Entanglement entropyof the eigenstates of H(2) for N = 12 and (inset shows N = 8). Note that thelow entanglement state at E = 0 is perhaps a consequence of the fact that thereare exponentially many states with E = 0. Data shown for PBC in the quantumnumber sector (k,X) = (0,+1).

H(p)− |Z2〉 = 0. The basis vectors of K(p)

+ are

∣∣∣F (p)j

⟩≡ 1√

c(p)j

(H(p)

+

)j ∣∣∣Z(p)2

⟩, j ≥ 0, (6.47)

where c(p)j is a normalization factor. The∣∣∣F (p)

j

⟩’s are all guaranteed to be orthogo-

nal since they have different charges. Indeed,∣∣∣F (p)

j

⟩has a chargeQ(p) = −Np2/2+

j because H(p)+ is a charge raising operator. Furthermore, the highest charge con-

figuration in K(p) can be shown to be is the configuration of Eq. (6.45) translated

by one unit cell, and it has a charge Q(p) = Np2/2 (resp. Q(p) = Np2/2 − p) for

PBC (resp. OBC), and thus K(p)+ is a Hilbert space of dimension D(p)

+ = (Np2+1)

(resp. D(p)+ = Np2 + 1− p).

The FSA is an approximation that K(p)+ is closed under the action of the total

231

Hamiltonian H(p). Since H(p) is of the form of Eq. (6.41) (i.e. a sum of charge

raising and lowering operators), using Eq. (6.47) we obtain

H(p)∣∣∣F (p)

j

⟩= β

(p)j+1

∣∣∣F (p)j+1

⟩+H(p)

−

∣∣∣F (p)j

⟩, (6.48)

where β(p)j =

√c(p)j /c

(p)j−1, where c(p) is the normalization factor defined in Eq. (6.47).

The crucial approximation of the FSA is thus

H(p)−

∣∣∣F (p)j

⟩≈ β

(p)j

∣∣∣F (p)j−1

⟩. (6.49)

While the approximation of Eq. (6.49) is only justified because of matching the

charge Q(p), we will show that this assumption leads to accurate predictions of the

energies of the quantum scars in this model. Thus, using Eqs. (6.48) and (6.49),

the Hamiltonian H(p) restricted to K(p)+ is a (D

(p)+ − 1)-dimensional tridiagonal

matrix in the FSA approximation that reads

H(p)FSA =

0 β(p)1 0 · · · · · · 0

β(p)1 0 β

(p)2 0

. . . ...

0 β(p)2

. . . . . . . . . ...... . . . . . . . . . . . . 0

... . . . . . . . . . 0 β(p)

D(p)+ −1

0 · · · · · · 0 β(p)

D(p)+ −1

0

. (6.50)

Indeed, the eigenstates of FSA Hamiltonian are known to reproduce the energies

of the scarred eigenstates of the PXP model to a good approximation. [66, 68]

These results are equivalent to those for the Hamiltonian H(1). We thus expect

232

0 20 40 60 80 100

T

0.0

0.2

0.4

0.6

0.8

1.0|⟨ (2

)2|e−i(H

+δH

) T|

(2)

2

⟩ |2

(a)

w

Tmin(w)

0.10 0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30

V1,0

0.10

0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

V2,0

(b)

5

10

15

20

25

30

35

40

45

Tmin

( w=

0.1)

Figure 6.5: (a) Fidelity of the |Z2 (k = 0)〉 state with (V1,0, V2,0) = (0.08, 0.04).(b) Late time fidelity of the |Z2 (k = 0)〉 state upon the addition of electrostaticterms. Clearly the line V1,0 = 2V2,0 has stronger revivals than other points inparameter space. Data shown for PBC and N = 16 in the quantum numbersector (k,X) = (0,+1).

that the eigenstates of H(p)FSA to be close to the quantum scars of H(p).

We numerically test these aspects for p = 2. We first define the momentum

k = 0 eigenstates constructed from the state∣∣∣Z(2)

2

⟩as

∣∣∣Z(2)2 (k = 0)

⟩≡

∣∣∣Z(2)2

⟩+ T

∣∣∣Z(2)2

⟩√2

, (6.51)

where T is the translation operator by one unit cell. Note that T 2∣∣∣Z(2)

2

⟩=∣∣∣Z(2)

2

⟩. In Fig. 6.3a, we plot the bipartite entanglement entropy S of the states

e−iH(2)T∣∣∣Z(2)

2 (k = 0)⟩

and e−iH(2)T∣∣R(2)

⟩states with the bipartition being one

half of the system containing N/2 consecutive unit cells. Note that while∣∣R(2)

⟩thermalizes quickly, the EE for the

∣∣∣Z(2)2 (k = 0)

⟩undergoes small oscillations be-

fore saturating close to the expected thermal value. Further, the EE growth of∣∣∣Z(2)2 (k = 0)

⟩is subballistic, contrary to the ballistic (linear in T ) growth observed

for∣∣R(2)

⟩, a characteristic of typical initial states for nonintegrable models. [238]

233

Furthermore, as shown in Fig. 6.3b, the fidelity of∣∣∣Z(2)

2 (k = 0)⟩

state, defined as∣∣∣⟨Z(2)2 (k = 0)

∣∣∣ e−iHT ∣∣∣Z(2)2 (k = 0)

⟩∣∣∣2, shows several strong revivals before oscillat-

ing and decaying at long times. As shown in Fig. 6.3c, the revivals survive up to

the largest system size (N = 20 unit cells) accessible via exact diagonalization. In

Fig. 6.4a, we plot the overlap of the eigenstates with the∣∣∣Z(2)

2 (k = 0)⟩

state, which

clearly show the existence of a “tower” of states approximately equally spaced in

energy that have a high overlap. Moreover, we compute H(2)FSA numerically and

show that the FSA (see Eqs. (6.49) and (6.50)) accurately predicts the energies of

these “scarred states”, providing evidence that the Krylov subspace K(2) is indeed

approximately closed under H(2). However, the low EE “scarred states” in the

spectrum of H(2) appear to strongly hybridize with the rest of the spectrum with

increasing N , as shown in Fig. 6.4b. While the hybridization can be attributed to

the fact that K(2)+ is only approximately closed under H(2), we do not understand

why the hybridization is weaker in the PXP (p = 1) model, although it takes

place there too [66, 68] and is likely to show a similar EE spectrum as Fig. 6.4b

for larger system sizes.

6.5.1 Stability of Scars

We now study the effect of perturbations of the scars obtained in Sec. 6.5. The full

quantum Hall spectrum exhibits level repulsion, [228] and presumably does not

exhibit scars. It is thus instructive to study how perturbations inspired from the

quantum Hall setup affect the scars obtained in Sec. 6.5. In this work, we restrict

ourselves to center-of-mass preserving electrostatic perturbations that occur in

a Landau level (see Eq. (E.38)). We consider the Hamiltonian H ′ = H + δH,

where H is the pair-hopping Hamiltonian of Eq. (5.1) and δH is a perturbation.

234

We consider the effect of nearest-neighbor and next-nearest-neighbor electrostatic

terms that appear in physical systems discussed in Appendix E. The perturbation

thus reads

δH =

Lb∑j=1

(V1,0nini+1 + V2,0nini+2). (6.52)

Since this is a diagonal perturbation, the Krylov subspaces K(p) are still closed

under the action of the perturbed Hamiltonian. When p = 1, using the mapping

of Eq. (6.12), the operator of Eq. (6.52) restricted to the subspace K(1) maps on

to the perturbation V1,0∑j

(1 + σzj )/2. The effect of this perturbation on the scars

of the PXP model were briefly studied in Ref. [68], where numerical evidence

suggested that the scars are stable upon the addition of small V1,0 < 1.

For general p, the electrostatic terms within the constrained subspace can be

written in terms of Zαj,n’s. To diagnose revivals in the presence of electrostatic

terms we use a measure Tmin (w), defined as the minimum time after which the

fidelity is always less than w, as depicted in Fig. 6.5a. We plot Tmin (w = 0.1) for

several values of V1,0 and V2,0 in Fig. 6.5b and observe that revivals are stable for

small strengths of electrostatic terms particularly when V1,0 = 2V2,0.

In contrast to electrostatic terms, certain longer range pair-hopping terms

generically do not preserve the Krylov subspaces K(p), invalidating the study of

the Hamiltonian H(p). For example, consider p = 1. The action of terms of C3,1 of

Eq. (E.38) c†j+1c†j+3cj+4cj and C3,2 of Eq. (E.38) c†jc

†j+5cj+3cj+2 for example read

c†j+1c†j+3cj+4cj | − o 〉 = | o − 〉 , c†jc

†j+5cj+3cj+2 | + − 〉 = | − + 〉 ,

(6.53)

configurations that are not allowed in K(1). We defer the detailed study of the

235

interplay of Krylov subspaces introduced by various long-range hopping terms for

future work.

6.6 Conclusions

We have studied the particular pair-hopping Hamiltonian introduced in Chap-

ter 6, a one dimensional model that arises within a Landau level in the thin-torus

limit of the Quantum Hall effect. At filling ν = 1/3, the pair-hopping Hamilto-

nian restricted to a particular constrained Krylov subspace space exactly maps

onto the PXP model, [66] which shows the existence of quantum many-body

scars [66, 68, 70] and exact strong ETH-violating eigenstates [75]. We showed

that this phenomenology generalizes to filling factors ν = p/(2p+1) (particularly

ν = 2/5), where “maximally squeezed” charge density wave configurations showed

revivals and subballistic growth of entanglement entropy in spite of strong hy-

bridization of the scarred eigenstates, contrary to the typical behavior in noninte-

grable models. Furthermore, we numerically explored the stability of the revivals

under electrostatic terms that appear in the quantum Hall Hamiltonian.

It is likely that similar phenomena occur with longer range pair-hopping terms

that arise in the quantum Hall setting and also multi-body hopping terms that

arise in the case of non-abelian quantum Hall states. It would be interesting to

understand the generic structure of such constrained Hilbert spaces at arbitrary

filling factors, whether or not they exhibit quantum scars, and if there is any con-

nection to quantum Hall physics. On the mathematical side, it would be inter-

esting to better understand constrained Hilbert spaces K(p) and the Hamiltonians

H(p). For example, the Hilbert space of the PXP model (and consequently K(1))

236

can be related to the configuration space of the Baxter Hard Square Model, which

gives rise to Bethe Ansatz integrable models that resemble the PXP model [71].

Moreover, the same Hilbert space K(1) arises in chains of Fibonacci anyons. An

interesting question is to explore any of these connections naturally generalize

to the Hilbert spaces K(p). On the physical side, an important challenge is to

identify physical interactions and regimes that naturally lead to the dominance of

pair-hopping terms in the quantum Hall Hamiltonian. This could provide a new

route to the experimental realization of quantum scars.

237

AAlgebra of dimers

A.1 Commutation relations

We start by defining operators that create and annihilate up (a†, a) or down (b†,

b) spin-1/2 Schiwinger bosons. By virtue of being bosons, they obey the boson

238

commutation relations as

[ai, a†j] = δij, [ai, aj] = 0, [bi, b

†j] = δij, [bi, bj] = 0, [a†i , bj] = 0, [ai, bj] = 0,

(A.1)

where i is the site index. We can then define dimer (singlet) annihilation and

creation operators between sites i and j as

cij = aibj − ajbi, c†ij = a†ib†j − a

†jb

†i . (A.2)

Since bosons on each site are identical, note that a singlet (antisymmetric) state

within each site vanishes (c†ii = 0). Using Eq. (A.1), one can derive the algebra

for the dimer operators as

[cmn, c†ij] = 2δmiδnj − 2δniδmj + (a†iam + b†ibm)δnj + (a†jan + b†jbn)δmi

−(a†ian + b†ibn)δmj − (a†jam + b†jbm)δni

[cmn, cij] = 0, [cij, c†ij] = 2 +Ni +Nj, (A.3)

where Ni = a†iai+ b†ibi, the total number operator of the Schwinger bosons on site

i. Commutation relations between the remaining operators can also be computed

as

[a†i , cmn] = −δimbn + δinbm, [b†i , cmn] = −δinam + δiman, [ai, cmn] = 0,

[bi, cmn] = 0, [Ni, c†mn] = (δin + δim)c

†mn, [Ni, a

†m] = δima

†m, [Ni, b

†m] = δimb

†m.(A.4)

239

Eqs. (A.1), (A.3), and (A.4) along with their Hermitian conjugates specify the

entire algebra of all the objects we are working with.

For our calculations, it is useful to know the expressions for the normal ordering

of the operator (c†ijcij)n. To compute this, we first need to know the commutation

relation [cij, (c†ij)

n]. Using the commutation relations in Eqs. (A.3) and (A.4), this

can be easily computed to be

[cij, (c†ij)

n] = (n(Ni +Nj) + n(3− n))(c†ij)n−1 (A.5)

We now work in a subspace of spin-S AKLT Hamiltonian basis states |ψS〉 that

satisfy Ni |ψS〉 = 2S |ψS〉 for any site i. Using this fact and Eq. (A.5), we can

expand (c†ijcij)n into a normal ordered form as

(c†ijcij)n =

n∑m=0

f(m,n)(c†ij)m(cij)

m. (A.6)

f(m,n) is determined recursively with the relations

f(n, n) = f(n− 1, n− 1), f(0, n) = δn,0

f(m,n) = f(m− 1, n− 1) + ((4S + 1)m−m2)f(m,n− 1). (A.7)

A.2 Dimer basis states and scattering rules for the

spin-1 AKLT model

In this section, we derive rules for action of the projector P (2,1)ij Eq. (1.6) on various

configurations of dimers around sites i and j, all of which are diagrammatically

240

21c c c cc c c cm m m m- - -m i j n

(a)

c c c c c cc c c c c cm m m m m m- -- -

m n ji p r(b)

FIG. 18. Two types of singlet configurations around a bondi, j.

Written diagrammatically, Eq. (C1) reads

| c c c- 6i j k

i + | c c c-6i j k

i = | c c c6-

i j ki . (C2)

Another example of linear dependence for dimer config-urations on four sites is, where i, j, k and l are distinct,

(c†ijc†kl + c†ilc

†jk) |i = ((a†i b

†j b†ia

†j)(a

†kb

†l b†ka

†l )

+ (a†i b†l b†ia

†l )(a

†jb

†k b†ja

†k)) |i

= (a†i b†k b†ia

†k)(a

†jb

†l b†ja

†l ) |i

= c†ikc†jl |i . (C3)

Written diagrammatically, Eq. (C3) reads

| c c c c- -i j k l

i + | c c c c--

i j k li = | c c c c- -

i j k li .(C4)

Appendix D: Dimer basis states and scattering rulesfor the spin-1 AKLT model

In this section, we derive rules for action of the pro-

jector P(2,1)ij Eq. (8) on various configurations of dimers

around sites i and j. Since the projector is normal or-dered, it is sucient to determine the action of (cij)m onthe dimer configurations. A useful identity in simplifyingdimer expressions is

cijc†mic

†jn |i = c†mn |i (D1)

where i, j,m, n are assumed to be distinct and |i, re-ferred to as the local vacuum, is a state annihilated byany of annihilation operators involving sites i, j,m, n.

1. Singlet basis states

Since each dimer has s = 0 and Sz = 0, any basisstate expressed in terms of only dimer creation operators(c†ijs) is a singlet state. For the spin-1 AKLTmodel, thereare two di↵erent possible configurations of dimers aroundtwo sites i and j. They are c†ijc

†mic

†jn |i (Fig. 18a) and

c†mic†nic

†jpc

†jr |i (Fig. 18b) where m,n, p, r are distinct

from i, j. Using Eq. (B3), we can derive the action ofcij on the two di↵erent kinds of dimer configurations.

cijc†ijc

†mic

†jn |i = c†ijc

†mn |i+ 4c†mic

†jn |i (D2)

cijc†mic

†nic

†jpc

†jr |i = c†mic

†npc

†jr |i c†nic

†mpc

†jr |i

c†mic†jpc

†nr |i c†nic

†jpc

†mr |i (D3)

c cc cm m6 66 6ji

(a)

c c cc c cm m m66 6-jin

(b)

c c c cc c c cm m m m- -6 6m i j n

(c)

c c c cc c c cm m m m--66

ji p r(d)

c c c c cc c c c cm m m m m-- -6

m ji p r(e)

c cc cm m6 6-ji

(f)

c c cc c cm m m6- -jim

(g)

FIG. 19. Types of non-singlet configurations around a bondi, j

Similarly, the expressions for the action of (cij)2 on theseconfigurations are

(cij)2c†ijc

†mic

†jn |i = 6c†mn |i (D4)

(cij)2c†mic

†nic

†jpc

†jr |i = 2(c†mpc

†nr + c†mrc

†np) |i .(D5)

Using Eqs. (D2) - (D5), and the expression for the pro-jector Eq. (8), the action of the projector on the config-urations can be written as

P(2,1)ij c†ijc

†mic

†jn |i = 0 (D6)

P(2,1)ij c†mic

†nic

†jpc

†jr |i = c†mic

†nic

†jpc

†jr |i

+ 1

4

c†ijc

†mic

†npc

†jr + c†ijc

†nic

†mpc

†jr

+c†ijc†mic

†jpc

†nr + c†ijc

†nic

†jpc

†mr

|i

+ 1

12

(c†ij)

2c†mrc†np + (c†ij)

2c†nrc†mp

|i (D7)

From Eq. (D6), the projector vanishes on any singletstate that contains a c†ij . This is heavily used in ourcalculations. As shown in Fig. 27, these can be writtenas a set of diagrammatic rules for obtaining a set of con-figurations into which an initial configuration of dimersscatters.

2. Non-singlet basis states

In the previous section we worked with singlet states,where all the configurations could be written onlyin terms of dimers. States with s 6= 0, becauseof SU(2) symmetry, would appear in multiplets of2s + 1 states. As mentioned in the main text, itis sucient to focus on the highest weight states ofeach multiplet. If s 6= 0, the configurations in thehighest weight multiplet would have free spin-" (a†i )

Figure A.1: Two types of singlet configurations around a bond i, j.

depicted in Fig. 1.12. Since the projector is normal ordered, it is sufficient to

determine the action of (cij)m on the dimer configurations. A useful identity in

simplifying dimer expressions is

cijc†mic

†jn |Θ〉 = −c†mn |Θ〉 (A.8)

where i, j,m, n are assumed to be distinct and |Θ〉, referred to as the local vacuum,

is a state annihilated by any of annihilation operators involving sites i, j,m, n.

A.2.1 Singlet basis states

Since each dimer has s = 0 and Sz = 0, any basis state expressed in terms of only

dimer creation operators (c†ijs) is a singlet state. For the spin-1 AKLT model, there

are two different possible configurations of dimers around two sites i and j. They

are c†ijc†mic

†jn |Θ〉 (Fig. A.1a) and c†mic

†nic

†jpc

†jr |Θ〉 (Fig. A.1b) where m,n, p, r are

distinct from i, j. Using Eq. (A.3), we can derive the action of cij on the two

different kinds of dimer configurations.

cijc†mic

†nic

†jpc

†jr |Θ〉 = −c

†mic

†npc

†jr |Θ〉 − c

†nic

†mpc

†jr |Θ〉 − c

†mic

†jpc

†nr |Θ〉 − c

†nic

†jpc

†mr |Θ〉

cijc†ijc

†mic

†jn |Θ〉 = −c

†ijc

†mn |Θ〉+ 4c†mic

†jn |Θ〉 (A.9)

241

21c c c cc c c cm m m m- - -m i j n

(a)

c c c c c cc c c c c cm m m m m m- -- -

m n ji p r(b)

FIG. 18. Two types of singlet configurations around a bondi, j.

Written diagrammatically, Eq. (C1) reads

| c c c- 6i j k

i + | c c c-6i j k

i = | c c c6-

i j ki . (C2)

Another example of linear dependence for dimer config-urations on four sites is, where i, j, k and l are distinct,

(c†ijc†kl + c†ilc

†jk) |i = ((a†i b

†j b†ia

†j)(a

†kb

†l b†ka

†l )

+ (a†i b†l b†ia

†l )(a

†jb

†k b†ja

†k)) |i

= (a†i b†k b†ia

†k)(a

†jb

†l b†ja

†l ) |i

= c†ikc†jl |i . (C3)

Written diagrammatically, Eq. (C3) reads

| c c c c- -i j k l

i + | c c c c--

i j k li = | c c c c- -

i j k li .(C4)

Appendix D: Dimer basis states and scattering rulesfor the spin-1 AKLT model

In this section, we derive rules for action of the pro-

jector P(2,1)ij Eq. (8) on various configurations of dimers

around sites i and j. Since the projector is normal or-dered, it is sucient to determine the action of (cij)m onthe dimer configurations. A useful identity in simplifyingdimer expressions is

cijc†mic

†jn |i = c†mn |i (D1)

where i, j,m, n are assumed to be distinct and |i, re-ferred to as the local vacuum, is a state annihilated byany of annihilation operators involving sites i, j,m, n.

1. Singlet basis states

Since each dimer has s = 0 and Sz = 0, any basisstate expressed in terms of only dimer creation operators(c†ijs) is a singlet state. For the spin-1 AKLTmodel, thereare two di↵erent possible configurations of dimers aroundtwo sites i and j. They are c†ijc

†mic

†jn |i (Fig. 18a) and

c†mic†nic

†jpc

†jr |i (Fig. 18b) where m,n, p, r are distinct

from i, j. Using Eq. (B3), we can derive the action ofcij on the two di↵erent kinds of dimer configurations.

cijc†ijc

†mic

†jn |i = c†ijc

†mn |i+ 4c†mic

†jn |i (D2)

cijc†mic

†nic

†jpc

†jr |i = c†mic

†npc

†jr |i c†nic

†mpc

†jr |i

c†mic†jpc

†nr |i c†nic

†jpc

†mr |i (D3)

c cc cm m6 66 6ji

(a)

c c cc c cm m m66 6-jin

(b)

c c c cc c c cm m m m- -6 6m i j n

(c)

c c c cc c c cm m m m--66

ji p r(d)

c c c c cc c c c cm m m m m-- -6

m ji p r(e)

c cc cm m6 6-ji

(f)

c c cc c cm m m6- -jim

(g)

FIG. 19. Types of non-singlet configurations around a bondi, j

Similarly, the expressions for the action of (cij)2 on theseconfigurations are

(cij)2c†ijc

†mic

†jn |i = 6c†mn |i (D4)

(cij)2c†mic

†nic

†jpc

†jr |i = 2(c†mpc

†nr + c†mrc

†np) |i .(D5)

Using Eqs. (D2) - (D5), and the expression for the pro-jector Eq. (8), the action of the projector on the config-urations can be written as

P(2,1)ij c†ijc

†mic

†jn |i = 0 (D6)

P(2,1)ij c†mic

†nic

†jpc

†jr |i = c†mic

†nic

†jpc

†jr |i

+ 1

4

c†ijc

†mic

†npc

†jr + c†ijc

†nic

†mpc

†jr

+c†ijc†mic

†jpc

†nr + c†ijc

†nic

†jpc

†mr

|i

+ 1

12

(c†ij)

2c†mrc†np + (c†ij)

2c†nrc†mp

|i (D7)

From Eq. (D6), the projector vanishes on any singletstate that contains a c†ij . This is heavily used in ourcalculations. As shown in Fig. 27, these can be writtenas a set of diagrammatic rules for obtaining a set of con-figurations into which an initial configuration of dimersscatters.

2. Non-singlet basis states

In the previous section we worked with singlet states,where all the configurations could be written onlyin terms of dimers. States with s 6= 0, becauseof SU(2) symmetry, would appear in multiplets of2s + 1 states. As mentioned in the main text, itis sucient to focus on the highest weight states ofeach multiplet. If s 6= 0, the configurations in thehighest weight multiplet would have free spin-" (a†i )

Figure A.2: Types of non-singlet configurations around a bond i, j

Similarly, the expressions for the action of (cij)2 on these configurations are

(cij)2c†ijc

†mic

†jn |Θ〉 = −6c†mn |Θ〉 , (cij)

2c†mic†nic

†jpc

†jr |Θ〉 = 2(c†mpc

†nr + c†mrc

†np) |Θ〉 .

(A.10)

Using the above equations, we obtain Eq. (D6) and (D7) in Fig. 1.12.

A.2.2 Non-singlet basis states

In the previous section we worked with singlet states, where all the configurations

could be written only in terms of dimers. States with s 6= 0, because of SU(2)

symmetry, would appear in multiplets of 2s+1 states. As discussed in Chapter 1,

it is sufficient to focus on the highest weight states of each multiplet. If s 6= 0,

the configurations in the highest weight multiplet would have free spin-↑ (a†i )

on the chain. Once this possibility is allowed, several new configurations are

possible. The distinct ones are a†i2a†j

2 |Θ〉 (Fig. A.2a), c†nia†ia

†j

2 |Θ〉 (Fig. A.2b),

242

c†nia†ia

†jc

†jp |Θ〉 (Fig. A.2c), (a†i )2c

†jpc

†jr |Θ〉 (Fig. A.2d), c†mia

†ic

†jpc

†jr |Θ〉 (Fig. A.2e),

c†ija†ia

†j |Θ〉 (Fig. A.2f), and c†ijc

†mia

†j |Θ〉 (Fig. A.2g), where m,n, p, r are distinct

from i, j. Analogous to Eq. (A.9), we derive identities for the action of cij on

each of these configurations.

cija†i

2a†j

2 |Θ〉 = 0, cijc†nia

†ia

†j

2 |Θ〉 = −2a†na†ia

†j |Θ〉

cijc†mia

†ia

†jc

†jn |Θ〉 = −(c

†mia

†ja

†n + a†ma

†ic

†jn + a†ia

†jc

†mn) |Θ〉

cij(a†i )

2c†jpc†jr |Θ〉 = −2(a

†ia

†rc

†jp + a†ia

†pc

†jr) |Θ〉

cijc†mia

†ic

†jpc

†jr |Θ〉 = −(c

†mic

†jpa

†r + c†mic

†jra

†p + c†mrc

†jpa

†i + c†mpc

†jra

†i ) |Θ〉

cijc†ija

†ia

†j |Θ〉 = 4a†ia

†j |Θ〉 , cijc

†ijc

†mia

†j |Θ〉 = −a†mc

†ij |Θ〉+ 4a†jc

†ij |Θ〉 .(A.11)

Similarly, the actions of (cij)2 are given by

(cij)2a†i

2a†j

2 |Θ〉 = 0, (cij)2c†nia

†ia

†j

2 |Θ〉 = 0, (cij)2c†mia

†ia

†jc

†jn |Θ〉 = 2a†ma

†n |Θ〉

(cij)2(a†i )

2c†jpc†jr |Θ〉 = 4a†pa

†r |Θ〉 , (cij)

2c†mia†ic

†jpc

†jr |Θ〉 = 2(c†mpa

†r + c†mra

†p) |Θ〉 ,

(cij)2c†ija

†ia

†j |Θ〉 = 0, (cij)

2c†ijc†mia

†j |Θ〉 = −6a†m |Θ〉 (A.12)

Again, using the expression of the projector in Eq. (1.6) and Eq. (A.12), we derive

Eqs. (D22)-(D28) in Fig. 1.12.

A.3 Dimer basis scattering rules for spin-S AKLT basis

states

In this section, we give a brief overview on how terms in the Hamiltonian Eq. (1.37)

act on basis states of a spin-S chain. To achieve this, since the Hamiltonian is

243

25

c c c c cs ss sc c c c cc c c c cc c c c c-- --

-

- -

-

ji

(a)

s ss sc c c c cc c c c cc c c c cc c c c c-- --

- -

-

ji

(b)

s s s ss ss s sc c c c cc c c c cc c c c cc c c c c- -6 66 66

-

ji

(c)

c c c cc c c cc c c cc c c css ss s s- --6 66 6ji

(d)

c c c cc c c cc c c cc c c c-s s sss s 6 666 66

-

ji

(e)

s s sss sc c c cc c c cc c c cc c c c66 66 66ji

(f)

FIG. 20. Scattering examples of S = 2 dimer configurationsunder the action of cij . Configurations of type (a) scatter toconfigurations of the type (b). Configurations such as (c) and(d) are annihilated by (cij)

4. Configuration (e) scatters to(f) under the action of (cij)

2. The configurations of the filledsmall circles are irrelevant for the scattering due to cij .

from i and j connected. For example, the spin-2 singletconfiguration shown in Fig. 20a scatters to configurationssuch as the one shown in Fig. 20b. If the basis state hasmany dimers (c†ij)

n, using Eqs. (B6) and (D2), the action

of cij gives rise to an additional term where the dimer c†ijis annihilated.With these observations, the action of cij on a singlet

basis state with no dimers c†ij and N dimers connectingeach of the sites i and j to other sites pl and rnrespectively, can be written as

cijNQl=1

c†pl,i

NQn=1

c†j,rn |i =

= NP

l0,n0=1

NQl=1,l 6=l0

c†pl,i

NQn=1,n 6=n0

c†j,rnc†l0,n0 |i (G1)

In Eq. (G1), on the first application of cij on the spin-Ssinglet state, N = 2S. Upon each application of cij , Ndecreases by 1 (one dimer is annihilated on each of thesites i and j). The action of (cij)m can be computed byconsecutive applications of Eq. (G1). The scattering con-figuration of (cij)m acting on the original spin-S singletstate would then be a sum of terms annihilating m pairsof dimers, each pair with one connecting site i and oneconnecting site j, and reconnecting the vacant sites in dif-ferent possible ways. The di↵erent ways to annihilate mdimer pairs that lead to the same scattering term resultin an overall m! factor. Morever, a factor of (1)m ap-pears because of the negative sign in Eq. (G1). Each c†ijin the original basis state gives rise to an additional scat-tering term where the c†ij is annihilated. Though tedious

to prove in general, we recover that P(J,S)

ij , for J > S

vanishes on any configuration containing (c†ij)S , similar

to Eq. (D6).To derive the non-singlet scattering rules, we need to

consider basis elements that consist of some free spin-"Schwinger bosons (a†). Firstly, cij annihilates any basisstate that does not have dimers on sites i or j, analo-gous to Eq. (D8). From Eqs. (D8) to (D14), observe thatthe action of cij on other basis states results in a sum ofterms with one free spin (or dimer, but not both spins)annihilated on each site i and j and the resulting vacantsite(s) populated with a free spin (dimer). As earlier,each c†ij in the original basis state gives rise to an ad-

ditional scattering term with the c†ij annihilated. Theaction for (cij)m can then be derived following the sameprocedure by repeated applications of cijs and account-ing for the overcounting factors.The important conclusion from the above observations

is that (cij)m annihilates all configurations on bondsi, j with a total of less thanm dimers on it or surround-ing it, analogous to Eqs. (D8), (D15), (D16) and (D20).For example, if a configuration has Ni dimers connectingsite i to a site di↵erent from j, Nj dimers connecting sitej to a site di↵erent from i and Nij dimers connectingsites i and j, the action of (cij)m can be written as

(cij)mNiQl=1

c†pl,i

NjQn=1

c†j,rn(a†i )

2SNi(c†ij)Nij (a†j)

2SNj |i = 0

if Ni +Nj +Nij < m (G2)

where the sites pl and rn are assumed to be dis-tinct from j and i respectively. For example, the spin-2configurations shown in Figs. 20c and 20d are annilatedby (cij)4 since they have a total of three dimers on andaround the bond i, j.Another useful configuration that we have used in our

calculations is if Ni+Nj = m and Nij = 0. As discussedabove, since each cij annilates a dimer connected to eitherof the sites i or j, up to an overall constant, the scatteringequation reads

(cij)mNiQl=1

c†pl,i

NjQn=1

c†j,rn(a†i )

2SNi(a†j)2SNj |i

(a†ia†j)

2SmNiQl=1

a†pl

NjQn=1

a†rn |i if Ni +Nj = m.

(G3)

This is analogous to Eqs. (D9), (D17) and (D18). InEq. (G3), all the dimers connected to sites i and j areannihilated. For example, the spin-2 configuration shownin Fig. 20e scatters to the one shown in Fig. 20f under theaction of (cij)2. Thus the term (c†ij)

m(cij)m in a Hamil-tonian acting on such a configuration results in a config-uration where the m dimers around bond i, j move onto bond i, j.

Figure A.3: Scattering examples of S = 2 dimer configurations under the actionof cij. Configurations of type (a) scatter to configurations of the type (b). Config-urations such as (c) and (d) are annihilated by (cij)

4. Configuration (e) scattersto (f) under the action of (cij)2. The configurations of the filled small circles areirrelevant for the scattering due to cij.

normal ordered, it is sufficient to compute the actions of (cij)m for m = S, . . . , 2S

on various configurations that can appear in a spin-S chain. However, such ex-

pressions are lengthy in general and here we merely note the structure of the

scattering terms. Note that the results in App. A.1 are valid for any value of S.

First we derive singlet scattering rules, analogous to the results in Appendix A.2.1.

If the basis element does not contain any c†ij, from Eqs. (A.8) and (A.9), the ac-

tion of cij on any basis state results in a sum of all possible configurations with

one dimer annihilated on each site i and j, and the resulting vacant sites on sites

different from i and j connected. For example, the spin-2 singlet configuration

shown in Fig. A.3a scatters to configurations such as the one shown in Fig. A.3b.

244

If the basis state has many dimers (c†ij)n, using Eqs. (A.5) and (A.10), the action

of cij gives rise to an additional term where the dimer c†ij is annihilated.

With these observations, the action of cij on a singlet basis state with no dimers

c†ij and N dimers connecting each of the sites i and j to other sites pl and rn

respectively, can be written as

cij

N∏l=1

c†pl,i

N∏n=1

c†j,rn |Θ〉 = −N∑

l′,n′=1

N∏l=1,l =l′

c†pl,i

N∏n=1,n =n′

c†j,rnc†l′,n′ |Θ〉 (A.13)

In Eq. (A.13), on the first application of cij on the spin-S singlet state, N =

2S. Upon each application of cij, N decreases by 1 (one dimer is annihilated on

each of the sites i and j). The action of (cij)m can be computed by consecutive

applications of Eq. (A.13). The scattering configuration of (cij)m acting on the

original spin-S singlet state would then be a sum of terms annihilating m pairs

of dimers, each pair with one connecting site i and one connecting site j, and

reconnecting the vacant sites in different possible ways. The different ways to

annihilate m dimer pairs that lead to the same scattering term result in an overall

m! factor. Morever, a factor of (−1)m appears because of the negative sign in

Eq. (A.13). Each c†ij in the original basis state gives rise to an additional scattering

term where the c†ij is annihilated. Though tedious to prove in general, we recover

that P (J,S)ij , for J > S vanishes on any configuration containing (c†ij)

S.

To derive the non-singlet scattering rules, we need to consider basis elements

that consist of some free spin-↑ Schwinger bosons (a†). Firstly, cij annihilates any

basis state that does not have dimers on sites i or j. In Eqs. (A.11), observe that

the action of cij on other basis states results in a sum of terms with one free spin

(or dimer, but not both spins) annihilated on each site i and j and the resulting

245

vacant site(s) populated with a free spin (dimer). As earlier, each c†ij in the original

basis state gives rise to an additional scattering term with the c†ij annihilated. The

action for (cij)m can then be derived following the same procedure by repeated

applications of cijs and accounting for the overcounting factors.

The important conclusion from the above observations is that (cij)m annihilates

all configurations on bonds i, j with a total of less than m dimers on it or

surrounding it. For example, if a configuration has Ni dimers connecting site i to

a site different from j, Nj dimers connecting site j to a site different from i and

Nij dimers connecting sites i and j, the action of (cij)m can be written as

(cij)m

Ni∏l=1

c†pl,i

Nj∏n=1

c†j,rn(a†i )

2S−Ni(c†ij)Nij(a†j)

2S−Nj |Θ〉 = 0 if Ni +Nj +Nij < m,

(A.14)

where the sites pl and rn are assumed to be distinct from j and i respectively.

For example, the spin-2 configurations shown in Figs. A.3c and A.3d are annilated

by (cij)4 since they have a total of three dimers on and around the bond i, j.

Another useful configuration that we have used in our calculations is if Ni+Nj =

m and Nij = 0. As discussed above, since each cij annilates a dimer connected to

either of the sites i or j, up to an overall constant, the scattering equation reads

(cij)m

Ni∏l=1

c†pl,i

Nj∏n=1

c†j,rn(a†i )

2S−Ni(a†j)2S−Nj |Θ〉 ∼ (a†ia

†j)

2S−mNi∏l=1

a†pl

Nj∏n=1

a†rn |Θ〉 if Ni+Nj = m.

(A.15)

In Eq. (A.15), all the dimers connected to sites i and j are annihilated. For

example, the spin-2 configuration shown in Fig. A.3e scatters to the one shown in

Fig. A.3f under the action of (cij)2. Thus the term (c†ij)m(cij)

m in a Hamiltonian

acting on such a configuration results in a configuration where the m dimers

246

around bond i, j move on to bond i, j.

247

BReview of Matrix Product Operators

B.1 Matrix Product Operators

We briefly review Matrix Product Operators (MPO), which are crucial to effi-

ciently act operators on a Matrix Product State (MPS) [108, 109, 239–241]. An

MPO representation of an operator O is defined as

O =∑

sn,tn

[blMTM

[s1t1]1 M

[s2t2]2 . . .M

[sLtL]L brM ] |sn〉 〈tn| . (B.1)

248

In Eq. (B.1), the operator O is written in terms of L χm × χm matrices with

elements expressed as d×d matrices acting on the physical indices. χm is referred

to as the bond dimension of the MPO and the corresponding vector space is the

auxiliary space. O can compactly be represented as a χm× χm× d× d tensor Mi

with two physical indices ([si], [ti] and two auxiliary indices. blM and brM are the

boundary vectors of the MPO in the operator auxiliary space.

Similar to an MPS, the construction of an MPO for a given operator is not

unique. We now describe a method to construct an MPO for an operator O. The

particular MPO construction we describe here relies on a generalized version of a

Finite State Automation (FSA). [108, 242, 243] An FSA is a system with a finite

set of “states” and a set of rules for transition between the states at each iteration.

In such a setup, each state maps to a unique state after an iteration. When the

states of the FSA are viewed as basis elements of a vector space, each state is

denoted as a vector and the transition between the states is described by a square

matrix. For example, we consider an FSA with two states |R〉 and |F 〉, that are

denoted as

|R〉 =

1

0

|F 〉 =

0

1

. (B.2)

If at each iteration, |R〉 and |F 〉 are interchanged, the transition matrix T is

T =

0 1

1 0

. (B.3)

In principle, these transition matrices could vary from an iteration to the next.

249

To exemplify the construction of an MPO, we start with a simple example:

O =L∑j=1

eikjCj (B.4)

where eikjCj can be written in the physical Hilbert space as

eikjCj ≡ eik1⊗ · · · ⊗ eik1︸ ︷︷ ︸j−1 times

⊗ eikC ⊗ 1⊗ · · · ⊗ 1︸ ︷︷ ︸L−j times

, (B.5)

such that the index j does not explicitly appear in any of the operators. Consider

an FSA that iterates L times and constructs the operator O by appending a

physical operator (either 1 or C) at each iteration to a string of operators. If |Sn〉

is the state of the FSA at the n-th iteration, the appended physical operator is

the matrix element 〈Sn|Tn |Sn+1〉 where Tn is the transition matrix at the n-th

iteration. For example, an FSA that constructs eikjCj of Eq. (B.5) starts in a

state |R〉. It remains in the state |R〉 for j− 1 iterations with a transition matrix

TR =

eik1 0

0 0

(B.6)

appending an 1 at each step. At the j-th iteration, the FSA transitions to |F 〉

(different from |R〉) with a transition matrix Tj

Tj =

0 eikC

0 0

, (B.7)

thus appending the operator C on site j and remains in |F 〉 in the rest of L − j

250

iterations with transition matrix

TF =

0 0

0 1

. (B.8)

O is then the sum of operators obtained using an FSA for all j. The sum over

operators can be efficiently represented by generalizing an FSA to allow for su-

perpositions of FSA states with operators as coefficients. For example, we allow

for FSA states such as eik1 |R〉 + eikC |F 〉. The transition matrix in such a gen-

eralized FSA is an arbitrary square matrix with operators as matrix elements.

Indeed, fixing the initial and final states of the FSA to be |R〉 and |F 〉, we can

construct the operator O with a transition matrix Mj on site j with elements:

Mj =

eik1 eikC

0 1

. (B.9)

Writing the entire process of the generalized FSA, 〈F |∏L

j=1Mj |R〉, we obtain

exactly the representation of O as an MPO of the form Eq. (B.1), where the

auxiliary space is the vector space spanned by states of the generalized FSA. Note

that since Mj does not depend on the site index j, we can omit this index. The

left and right boundary vectors blM and brM are the vector representations of the

FSA states |R〉 and |F 〉 respectively (Eq. (B.2)),

blM =

1

0

brM =

0

1

. (B.10)

The MPO representations of more general operators can be computed similarly

251

with the introduction of intermediate states of the generalized FSA. For example,

in the construction of the MPO for the operator

O =∑j

eikj(WjXj+1

), (B.11)

one introduces an intermediate state |I1〉 of the generalized FSA, such that the

transition matrix elements at any step read 〈R|T |I1〉 = eikW and 〈I1|T |F 〉 = X.

The MPO for O in the auxiliary dimension thus reads

M =

eik1 eikW 0

0 0 X

0 0 1

. (B.12)

The bond dimension of the MPO χm is the number of states of the generalized

FSA generating it. Since the initial state of the FSA is |R〉 and the final state is

|F 〉, the components of the left and right boundary vectors of an MPO are always

(blM)α = δα,1 (brM)α = δα,χm (B.13)

Since the flow of an FSA is uni-directional, the MPO is always an upper tri-

angular matrix in the auxiliary indices. For a translation invariant MPO, any

element on the MPO diagonal appears in the operator as multiple direct products

of the same operator. For example, the MPO

MO =

W C

0 X

(B.14)

252

represents an operator O defined on a lattice of length L that reads

O =

(L−1∏i=1

Wi

)CL + C1

(L∏i=2

Xi

)+ . . . , (B.15)

which is not a strict local operator unless W and X are proportional to 1. Thus,

for an operator that is a sum of strictly local terms, the only diagonal element that

can appear in the MPO is 1, up to an overall constant (such as eik). Moreover, if

the diagonal element in an MPO corresponding to an intermediate state is 1, the

operator O includes a non-local term, i.e. a long range coupling between sites.

For example,

MO =

1 W 0

0 1 X

0 0 1

=⇒ O =L−1∑i=1

L∑j=i+1

WiXj. (B.16)

Thus, for operators that are the sum of non-trivial operators with a finite support,

the only non-vanishing diagonal elements correspond to the auxiliary states |R〉

and |F 〉.

B.2 Jordan normal form of block upper triangular ma-

trices

Due to the upper-triangular structure of MPOs, the transfer matrices of an MPO

× MPS discussed in Chapter 2 is typically a block upper triangular matrix. In

this section we describe a systematic procedure to determine the structure of

generalized eigenvalues, eigenvectors and Jordan normal forms of particular block

253

upper triangular matrices that arise in the analysis of the MPO × MPS states in

the context of MPO × MPS, which are of the form

M =

M11 M12 M13 . . . M1D

0 M22 M23. . . M2D

... . . . . . . . . . ...

... . . . . . . MD−1,D−1 MD−1,D

0 . . . . . . 0 MDD

(B.17)

where diagonal submatrices Mii’s are χ× χ diagonalizable matrices that have at

most a single non-degenerate eigenvalue of magnitude 1. We assume d of the

diagonal submatrices have an eigenvalue of magnitude 1, and they are written as

Mσ(i),σ(i), 1 ≤ i ≤ d, where

σ : 1, . . . , d → 1, . . . , D

σ(i) = j =⇒ Mjj is the i’th block with eigenvalue of magnitude 1.(B.18)

Furthermore, we restrict ourselves to determining the Jordan block structure of

generalized eigenvalues of unit magnitude and the structure of the corresponding

generalized eigenvectors.

We first derive the generalized eigenvalues of M using its characteristic equa-

tion. Note that for any λ,

det(M − λ1Dχ) =D∏i=1

det(Mii − λ1χ). (B.19)

Thus, the generalized eigenvalues of M are the eigenvalues of its submatrices on

254

the diagonal. However, as we will see, an eigenvector of M corresponding to

an eigenvalue λα need not exist, particularly due to the upper triangular struc-

ture of M . In such a case, M is not diagonalizable, λα is called a generalized

eigenvalue, and corresponding generalized eigenvector exists. In general, a Jordan

decomposition of M of the form

M = PJP -1 (B.20)

always exists, where J is the Jordan normal form of M , the columns of P are

the right generalized eigenvectors of M and the rows of P -1 are its left general-

ized eigenvectors. Since P -1P = 1Dχ, the conventional form for the generalized

eigenvectors of M is

lTαrβ = δαβ (B.21)

where lα and rβ are left and right generalized eigenvectors of M , the rows and

columns of P -1 and P respectively. We now derive the form of lα and rβ when

M has the form of Eq. (B.17).

The Jordan normal form J of M is related to M by means of a similarity

transformation, that is,

J = P -1MP . (B.22)

Thus, we can construct J , P and P -1 by sequentially performing similarity trans-

formations on M to reduce it to a Jordan normal form. A similarity transforma-

tion on a matrix B using a matrix A is defined as the transformation

B → A-1BA. (B.23)

255

Before we show the explicit construction of the Jordan normal form, we summarize

the three main steps that we use to proceed:

(I) A similarity transformation of M using a block-diagonal matrix ∆. The

resultant matrix is Λ(1,2),

Λ(1,2) = ∆-1M∆. (B.24)

Λ(1,2) has the form

Λ(1,2) =

Λ11 Λ(1,2)12 Λ

(1,2)13 . . . Λ

(1,2)1D

0 Λ22 Λ(1,2)23

. . . Λ(1,2)2D

... . . . . . . . . . ...

... . . . . . . ΛD−1,D−1 Λ(1,2)D−1,D

0 . . . . . . 0 ΛDD

, (B.25)

where Λii is the eigenvalue matrix of Mii.

(II) A similarity transformation is then applied to Λ(1,2) using a carefully chosen

block-upper triangular matrix O, such that

Λ = O-1Λ(1,2)O, (B.26)

256

where Λ can be written as

Λ =

Λ11 Λ12 Λ13 . . . Λ1D

0 Λ22 Λ23. . . Λ2D

... . . . . . . . . . ...

... . . . . . . ΛD−1,D−1 ΛD−1,D

0 . . . . . . 0 ΛDD

(B.27)

where

(Λij)αβ 6= 0 =⇒ (Λii)αα = (Λjj)ββ , i < j. (B.28)

O in Eq. (B.26) has the form

O =D∏j=2

(1∏

i=j−1

Oij

), (B.29)

where Oij and O-1ij respectively read

Oij =

1 0 · · · · · · · · · 0

0. . . . . . . . . . . . ...

... . . . . . . Oij. . . ...

... . . . . . . . . . . . . ...

... . . . . . . . . . . . . 00 · · · · · · · · · 0 1

i rows

︸ ︷︷ ︸j columns

O-1ij =

1 0 · · · · · · · · · 0

0. . . . . . . . . . . . ...

... . . . . . . −Oij. . . ...

... . . . . . . . . . . . . ...

... . . . . . . . . . . . . 00 · · · · · · · · · 0 1

i rows

︸ ︷︷ ︸j columns

.(B.30)

(III) A similarity transformation S of the form is applied to Λ to obtain the

257

Jordan normal form J , such that

J = S-1ΛS. (B.31)

Going through steps (I)-(III) above, we find that the left and right generalized

eigenvectors corresponding to generalized eigenvalues of unit magnitude generi-

cally have the following forms:

rj =

∗...

∗

cjrj

0

...

0

, lj =

0

...

0

ljcj

∗...

∗

, (B.32)

where rj and lj are the left and right eigenvectors of Mjj corresponding to eigen-

value of unit magnitude, and cj is a non-zero constant that need not be the same

as sj since lj and rj can be rescaled freely in a way that lTj rj = 1. For a detailed

construction of block upper triangular matrix in complete generality, we refer to

discussions in Refs. [244, 245].

258

CEmbedding Quasiparticles using MPS

Subspaces

259

C.1 Total Angular Momentum Eigenstates

In this section, we list the various total angular momentum eigenstates of two

spin-1’s. We denote the single site spin-1 basis vectors with Sz = +1, 0,−1 by

|+〉 , |0〉 , |−〉 respectively. Labelling the state with total spin j, j ∈ 0, 1, 2 and

its z-projection m, m ∈ −j,−j + 1, · · · , j as |Jj,m〉, they read

|J2,±2〉 = |± ±〉 , |J2,0〉 = 1√6(|+ −〉+ 2 |0 0〉+ |− +〉) , |J2,±1〉 = 1√

2(|± 0〉+ |0 ±〉)

|J1,±1〉 = 1√2(|± 0〉 − |0 ±〉) , |J1,0〉 = 1√

2(|+ −〉 − |− +〉)

|J0,0〉 = 1√3(|+ −〉 − |0 0〉+ |− +〉) . (C.1)

C.2 Examples of A and B subspaces for the AKLT-like

MPS

Here we compute the subspaces A and B of Eqs. (3.16) and (3.25) for an MPS of

the form

A[+] = c+σ+, A[0] = c0σ

0, A[−] = c−σ−, (C.2)

where σ+, σ−, and σ0 ≡ σz are the Pauli matrices. Note that the AKLT MPS of

Eq. (2.1) is recovered by setting

(c+, c0, c−) =1√3

(√2,−1,−

√2). (C.3)

260

Compactly, we can write A[m] = cmσm. We first compute the subspace A (defined

in Eq. (3.16)) for the two-site MPS

A = spanX

∑m,n∈+,0,−

cmcnTr [Xσmσn] |m,n〉

. (C.4)

Choosing X from the convenient basis 1√21, σ+, 1√

2σz, σ− of 2× 2 matrices, the

subspace A in Eq. (C.4) can be straightforwardly computed to be (after normal-

ization)

A = span 1√2(|c+c−|2+2|c0|4)

(c+c− (|+−〉+ |−+〉) + 2c20 |00〉) ,

1√2(|−0〉 − |0−〉) , 1√

2(|+−〉 − |−+〉) , 1√

2(|+0〉 − |0+〉), (C.5)

Thus, for the AKLT MPS, we obtain

A = span|J0,0〉 , |J1,−1〉 , |J1,0〉 , |J1,1〉, (C.6)

where the total angular momentum eigenstates |Jj,m〉 are enumerated in App. C.1.

Using the operator O = (S+)2, the B subspace defined in Eq. (3.26) can be directly

computed to be

B = span|J2,1〉 , |J2,2〉, (C.7)

which is independent of the cm’s. Thus, using Eqs. (C.5) and (C.7), we obtain

Ac/B = span|J2,−1〉 , |J2,−2〉 ,1√

|c+c−|2 + 2|c0|4(−c20 (|+−〉+ |−+〉) + c+c− |00〉

).

(C.8)

261

C.3 Single-Site Quasiparticle Exact Eigenstates in the

MPS Language

C.3.1 Single quasiparticle

In this section we show that the conditions of Eqs. (3.14) and (3.22) imply the

existence of a quasiparticle eigenstate of the Hamiltonian Eq. (3.13). Rewriting

the conditions here for convenience, they read

hj |AA〉 = 0 (C.9)(hj − E

) (|BA〉+ eik |AB〉

)= 0. (C.10)

Note that in Eqs. (C.9) and (C.10), hj is a two-site operator, with j being the left

site. Using these two expressions, the action of the Hamiltonian H on the state

with one quasiparticle reads

Hj

|[A · · ·ABA · · ·A]〉 =(hj−1 + hj

) j

|[A · · ·ABA · · ·A]〉

=(hj−1 + E

) j

|[A · · ·ABA · · ·A]〉 − eik(hj − E

) j

|[A · · ·AAB · · ·A]〉, (C.11)

where the first line follows from Eq. (3.14) and the second line from Eq. (3.22)

with hj. Defining the shorthand notation

|Bj〉 ≡j

|[A · · ·ABA · · ·A]〉, (C.12)

262

Eq. (C.11) can be rewritten as

H |Bj〉 =(hj−1 + E

)|Bj〉 − eik

(hj − E

)|Bj+1〉 . (C.13)

Thus,

H |ψA (B, k)〉 = HL∑j=1

eikj |Bj〉 =L∑j=1

eikj[(hj−1 |Bj〉 − eikhj |Bj+1〉

)+ E

(|Bj〉+ eik |Bj+1〉

)]=

L∑j=1

eikj[(hj−1 |Bj〉 − hj−1 |Bj〉

)+ E (|Bj〉+ |Bj〉)

]= 2E

L∑j=1

eikj |Bj〉 = 2E |ψA (B, k)〉 .(C.14)

Thus, the conditions of Eqs. (3.14) and (3.22) guarantee a quasiparticle eigenstate

of H with energy E = 2E .

C.3.2 Tower of states

Here we show that in addition to Eqs. (3.14) and (3.22), Eqs. (3.30) and (3.31)

guarantee the existence of a tower of quasiparticle exact eigenstates (rewriting

here for convenience)

O2 |B〉 = 0, |BB〉 = 0. (C.15)

We first illustrate the exactness for two quasiparticles dispersing in the ground

state background, by defining the configuration of two quasiparticles as

|Bj1 , Bj2〉 =j1 j2

|[A · · ·ABA · · ·ABA · · ·A]〉. (C.16)

Note that |Bj, Bj+1〉 = 0 and |Bj, Bj〉 = 0. As a consequence of Eq. (C.15), we

are guaranteed to have at least one A in between the B’s in the configuration of

263

Eq. (C.16). Thus, the Hamiltonian H acts independently on each of the quasi-

particles. That is, similar to Eq. (C.13), we obtain (with subscripts taken modulo

L)

H |Bj, Bj+n〉 =

(hj−1 + E + hj+n−1 + E

)|Bj, Bj+n〉

−eik(hj − E

)|Bj+1, Bj+n〉 − eik

(hj+n − E

)|Bj, Bj+n+1〉 if 3 ≤ n ≤ L− 3(

hj−1 + E + hj+1 + E)|Bj, Bj+2〉

−eik(hj+2 − E

)|Bj, Bj+3〉 if n = 2(

hj−1 + E + hj+L−3 + E)|Bj, Bj+L−2〉

−eik(hj − E

)|Bj+1, Bj+L−2〉 if n = L− 2

0 otherwise

.

(C.17)

To write Eq. (C.17) compactly, we first obtain a useful identity:

(hj+1 − E

)|Bj, Bj+2〉 =

(hj+1 − E

) j j+2

|[A · · ·ABABA · · ·A]〉 =

−e−ik(hj+1 − E

) j j+2

|[A · · ·ABBAA · · ·A]〉 = 0. (C.18)

Note that Eq. (C.19) can be written as

H |Bj1 , Bj2〉 =(hj1−1 + E + hj2−1 + E

)|Bj1 , Bj2〉 − eik

(hj1 − E

)|Bj1+1, Bj2〉

−eik(hj2 − E

)|Bj1 , Bj2+1〉 ∀j1, j2, (C.19)

264

where we have used Eq. (C.18). Using Eq. (C.19), we obtain that

HL∑

j1,j2=1

eik(j1+j2) |Bj1 , Bj2〉 =L∑

j1,j2=1

eik(j1+j2)[(hj1−1 + E

)|Bj1 , Bj2〉 − eik

(hj1 − E

)|Bj1+1, Bj2〉

+(hj2−1 + E

)|Bj1 , Bj2〉 − eik

(hj2 − E

)|Bj1 , Bj2+1〉

]=

L∑j1,j2=1

eik(j1+j2)[(hj1−1 + E

)|Bj1 , Bj2〉 − eik

(hj1 − E

)|Bj1+1, Bj2〉

]+

L∑j1,j2=1

[eik(j1+j2)

(hj2−1 + E

)|Bj1 , Bj2〉 − eik

(hj2 − E

)|Bj1 , Bj2+1〉

]= 2

L∑j1,j2=1

eik(j1+j2)(hj1−1 + E

)|Bj1 , Bj2〉 − 2

L∑j1,j2=1

eik(j1+1+j2)(hj1 − E

)|Bj1+1, Bj2〉

= 4EL∑

j1,j2=1

eik(j1+j2) |Bj1 , Bj2〉, (C.20)

where in the third step we have interchanged j1 and j2 in the second sum. Sim-

ilarly, we obtain an exact eigenstate with n quasiparticles of momentum k, pro-

vided the quasiparticles are constrained to be separated by at least one site. Thus,

we obtain a quasiparticle tower of exact eigenstates |S2n〉 with energies 2nE.

Note that this tower of eigenstates can consist of identical quasiparticles of any

momentum k provided there exists a tensor B for which Eqs. (C.9), (C.10), (C.15)

are satisfied.

C.4 SU(2) Multiplet of the Spin-2 Magnon for the AKLT

chain

We obtain the quasiparticle creation operators for the spin-2 magnon multiplet of

the AKLT chain. Representing the AKLT ground state as |G〉, the highest weight

state of the spin-2 magnon exact eigenstate (up to a normalization constant) reads

265

|S2〉 ≡ P |G〉 =L∑j=1

(−1)j(S+j )

2 |G〉 . (C.21)

Since the AKLT Hamiltonian is SU(2) symmetric [55] and |S2〉 has a total spin

2, we can obtain 5 linearly independent eigenstates with the same energy. They

are

|S2〉 , S− |S2〉 , (S−)2 |S2〉 , (S−)3 |S2〉 , (S−)4 |S2〉, (C.22)

where S− is the lowering operator S− ≡L∑j=1

S−j .

We now express the rest of the states in the multiplet of Eq. (C.22) as quasi-

particles states of the form of Eq. (3.39). Note that the commutation relations of

onsite spin-1 operators read

[S+j , S

−k ] = Szj δj,k, [Szj , S

±k ] = ±S

±j δj,k. (C.23)

We start with the Sz = +1 state and using the fact that S− |G〉 = 0, we express

it as

S− |S2〉 = [S−,P ] |G〉 =L∑j=1

(−1)j[S−j , (S

+j )

2] |G〉 ∝L∑j=1

(−1)jSzj , S+j |G〉 ,

(C.24)

where we have used Eq. (C.23) and omitted an overall normalization factor. Sim-

ilarly, we can apply the lowering operator on Eq. (C.24) and write

(S−)2 |S2〉 =L∑j=1

(−1)j[S−j , Szj , S+

j ] |G〉 ∝L∑j=1

(−1)j(2(Szj )2 − S+j , S

−j )) |G〉 ,

(C.25)

266

Repeating the same steps again, we also obtain

(S−)3 |S2〉 =L∑j=1

(−1)jS−j , S

zj |G〉 , (S−)4 |S2〉 =

L∑j=1

(−1)j(S−j )

2 |G〉 . (C.26)

Hence, an arbitrary eigenstate in the multiplet of the spin-2 magnon exact eigen-

state is given by

|ψ〉 =L∑j=1

(−1)jOj |G〉 , O ∈ span(S+)2, Sz, S+, 2(Sz)2−S+, S−, S−, Sz, (S−)2.

(C.27)

C.5 Examples of A and B subspaces for the Potts-like

MPS

Here we present the derivation of the subspaces A and B of Eqs. (3.16) and (3.56)

for an MPS of the form

A[+] = c+σ+, A[0] = c012×2 ≡ c0σ

0, A[−] = c−σ−, (C.28)

where σ+, and σ− are the Pauli matrices. Note that the Potts MPS of Eq. (3.67)

is recovered by setting

(c+, c0, c−) =1√2(1, 1, 1) . (C.29)

267

The computation of the A proceeds similarly to that for the generalized AKLT

MPS in App. C.2. Using the MPS matrices of Eq. (C.28) instead, we obtain

A = span 1√2(|c+c−|2+2|c0|4)

(c+c− (|+−〉+ |−+〉) + 2c20 |00〉) ,

1√2(|−0〉+ |0−〉) , 1√

2(|+−〉 − |−+〉) , 1√

2(|+0〉+ |0+〉). (C.30)

Thus, for the perturbed Potts MPS, using Eq. (C.29)

A = span|J2,0〉 , |J2,−1〉 , |J1,0〉 , |J2,1〉, (C.31)

where the total angular momentum eigenstates |Jj,m〉 are enumerated in App. C.1.

We then use O(2) = (S+)2⊗S++S+⊗ (S+)2 and compute the B subspace defined

in Eq. (3.56) to be

B = |J2,2〉 (C.32)

which is independent of the cm’s. Using Eqs. (C.30) and (C.32), we obtain

(Ac/B) = span|J2,−2〉 , |J1,−1〉 , |J1,1〉 ,1√

|c+c−|2 + 2|c0|4(−c20 (|+−〉+ |−+〉) + c+c− |00〉

).

(C.33)

268

DEta pairing and RSGAs

D.1 Useful Identities

In this appendix, we provide some useful operator identities that we use in this

article. We denote spinful fermionic creation and annihilation operators by c†r,σ

and cr,σ, where r denotes the site index and σ the spin index. These obey the

algebra

cr,σ, cr′,σ′ = c†r,σ, c†r′,σ′ = 0, cr,σ, c†r′,σ′ = δr,r′δσ,σ′ . (D.1)

269

Further, defining the operators

nr,σ ≡ c†r,σcr,σ, η†r ≡ c†r,↑c†r,↓, ηr ≡ −cr,↑cr,↓ (D.2)

we directly obtain the useful relations

[c†r,σ, nr′,σ′ ] = −δr,r′δσ,σ′c†r,σ, [cr,σ, nr′,σ′ ] = δr,r′δσ,σ′cr,σ. (D.3)

We also obtain

[nr,σ, η†r′ ] = δr,r′η†r, [nr,↑nr,↓, η

†r′ ] = δr,r′η†r, (D.4)

[η†r′ , η†r] = 0, [ηr′ , η†r] = δr,r′(1− nr,↑ − nr,↓). (D.5)

Using Eq. (D.3), we also obtain

[c†r′,σ′cr,σ, η†r] = −sσc

†r,σc

†r′,σ′ , sσ =

+1 if σ =↑

−1 if σ =↓, σ =

↓ if σ =↑

↑ if σ =↓.

(D.6)

D.2 η-pairing with disorder and spin-orbit coupling

Here we derive the conditions for the η operator of Eq. (4.8) to commute with a

generic one-body hopping operator of the form

T σ,σ′

r,r′ =(tσ,σ

′

r,r′c†r,σcr′,σ′ + tσ

′,σr′,rc

†r′,σ′cr,σ

). (D.7)

270

Our aim is to determine a set of conditions on qr, tσ,σ′

r,r′ such that

[∑σ,σ′

T σ,σ′

r,r′ , η†] = [

∑σ,σ′

T σ,σ′

r,r′ , qrη†r + qr′η†r′ ] = 0. (D.8)

We first compute [T σ,σ′

r,r′ , qrη†r]:

[T σ,σ′

r,r′ , qrη†r] = [tσ,σ

′

r,r′c†r,σcr′,σ′ + tσ

′,σr′,rc

†r′,σ′cr,σ, qrc

†r,↑c

†r,↓] = −qrt

σ′,σr′,rsσc

†r,σc

†r′,σ′ ,

[T σ,σ′

r,r′ , qr′η†r′ ] = [tσ,σ′

r,r′c†r,σcr′,σ′ + tσ

′,σr′,rc

†r′,σ′cr,σ, qr′c†r′,↑c

†r′,↓] = −qr′tσ,σ

′

r,r′sσ′c†r′,σ′c

†r,σ(D.9)

where we have used Eqs. (D.1) and (D.6). Using Eq. (D.8), we obtain

[∑σ,σ′

T σ,σ′

r,r′ , η†] = −

∑σ,σ′

(qrt

σ′,σr′,rsσ − qr′tσ,σ

′

r,r′sσ′

)c†r,σc

†r′,σ′ = −

∑σ,σ′

(qrt

σ′,σr′,rsσ + qr′tσ,σ

′

r,r′sσ′

)c†r,σc

†r′,σ′ ,

(D.10)

where we have used Eq. (D.1) and sσ′ = −sσ′ . Eq. (D.8) is thus satisfied by

setting

qrtσ′,σr′,rsσ + qr′tσ,σ

′

r,r′sσ′ = 0 ∀σ, σ′. (D.11)

D.3 Tower of States from (Restricted) Spectrum Gen-

erating Algebras

Here we show that the (Restricted) Spectrum Generating Algebras lead to the

existence of a tower of exact eigenstates of the Hamiltonian. We work with a

Hamiltonian H and “root eigenstate” |ψ0〉 from which the tower is generated by

the application of η† operator. We define a set of states |ψn〉 as |ψn〉 ≡ (η†)n |ψ0〉,

and a set of operators Hn as H0 ≡ H, and Hn+1 ≡ [Hn, η†].

271

Lemma D.3.1 (SGA). If the Hamiltonian H and operator η† satisfy the condi-

tions

(i) H |ψ0〉 = E0 |ψ0〉 (ii) [H, η†] = Eη† (D.12)

then

H |ψn〉 = (E0 + nE) |ψn〉 or |ψn〉 = 0. (D.13)

Proof. The proof proceeds straightforwardly via induction. Assuming |ψm〉 satis-

fies Eq. (D.13), we show |ψm+1〉 satisfies Eq. (D.13) provided it does not vanish.

Using condition (ii), we obtain

[H, η†] |ψm〉 = E |ψm〉 =⇒ (Hη† − η†H) |ψm〉 = E |ψm〉

=⇒ Hη† |ψm〉 = (E0 +mE + E) η† |ψm〉 . (D.14)

Thus, either |ψm+1〉 = 0 or H |ψm+1〉 = (E0 + (m+ 1)E) |ψm+1〉. Since Eq. (D.13)

is satisfied for m = 0 (due to condition (i)), this concludes the proof.

Lemma D.3.2 (RSGA-1). If the Hamiltonian H, operator η†, and state |ψ0〉 such

that η† |ψ0〉 6= 0 satisfy the conditions

(i) H |ψ0〉 = E0 |ψ0〉 (ii) [H, η†] |ψ0〉 = Eη† |ψ0〉 (iii) [[H, η†], η†] = 0 (D.15)

then

H |ψn〉 = (E0 + nE) |ψn〉 or |ψn〉 = 0. (D.16)

Proof. The proof proceeds by induction on two levels. We first wish to show

H1 |ψn〉 = E |ψn+1〉 (D.17)

272

For the purposes of induction, we assume Eq. (D.17) is valid for |ψm〉. Using

condition (iii) we obtain

H2 |ψm〉 = [H1, η†] |ψm〉 = 0 =⇒ (H1η

†−η†H1) |ψm〉 = 0 =⇒ H1 |ψm+1〉 = E |ψm+2〉 .

(D.18)

Since Eq. (D.17) is satisfied for m = 0 (due to condition (ii)), this concludes the

proof of Eq. (D.17).

Using Eq. (D.17), we show Eq. (D.16) by induction again. Assuming Eq. (D.16)

holds for |ψm〉, using Eq. (D.14) we can show |ψm+1〉 also satisfies it. Since |ψ0〉

satisfies Eq. (D.16) (due to condition (i)), this concludes the proof.

Lemma D.3.3 (RSGA-M). If the Hamiltonian H, operator η†, and state |ψ0〉

such that (η†)n |ψ0〉 6= 0 for n ≤M satisfy the conditions

(i) H |ψ0〉 = E0 |ψ0〉 (ii) H1 |ψ0〉 = E |ψ1〉 (iii) Hn |ψ0〉 = 0 ∀n, 2 ≤ n ≤M

(iv) HM+1 = 0 (i.e. [HM , η†] = 0) (D.19)

then

H |ψn〉 = (E0 + nE) |ψn〉 or |ψn〉 = 0. (D.20)

Proof. We start with conditions (iii) and (iv) and note that they satisfy the con-

ditions (i) and (ii) of Lemma D.3.1 with the replacements H → HM , E0 → 0, and

E → 0. Using Eq. (D.13), we arrive at

HM |ψn〉 = 0 = [HM−1, η†] |ψn〉 ∀ n. (D.21)

Further, using Eq. (D.14) along with condition (i) of Lemma D.3.1 with the re-

273

placements H → HM−1, E0 → 0, E → 0, as a consequence of Eq. (D.21) we obtain

HM−1 |ψn〉 = 0 = [HM−2, η†] |ψn〉 ∀ n, (D.22)

which is the same as Eq. (D.21) with the replacements HM → HM−1 and HM−1 →

HM . Repeating the steps from Eq. (D.21) to Eq. (D.22) successively replacing

Hn → Hn−1 at each step, we finally arrive at H2 |ψn〉 = 0 = [H1, η†] |ψn〉 forall n.

The proof of Eq. (D.20) can then be completed following the same steps as the

proof of Lemma D.3.2.

274

E

275

Physical Origins of the Pair-Hopping

Hamiltonian

E.1 Bloch Many-Body Localization

E.1.1 Schrieffer-Wolff Transformation for the Bloch MBL Hamil-

tonian

We explicitly derive the pair-hopping Hamiltonian Eq. (5.1) in the large E/t limit

of the Bloch MBL Hamiltonian:

HBloch = t∑j

(c†jcj+1 + h.c.

)+ E

∑j

j nj + V0∑j

nj + V1∑j

wjnjnj+1, (E.1)

where we have omitted the limits on the sums since we consider a chain of infinite

length. Furthermore, we treat t, V0, V1 perturbatively and hence rescale Eq. (E.1)

by E, such that the Hamiltonian is recast as

H ≡ HBloch

E= C + λ

(T+ + T− + V

), (E.2)

276

where C is the CoM operator (for OBC) C =∑

j jnj, T = T+ + T− and V =

V0 + V1, with

T+ =∑j

c†j+1cj, T− =∑j

c†jcj+1 = T †+, V0 = α0

∑j

wjnj, V1 = α1

∑j

njnj+1.

(E.3)

Here, the parameters are defined as λ = tE

and αν = Vνt

, for ν ∈ 0, 1 and where

we work in the regime where ασ ∼ O (1).

As is clear from Eq. (E.3), T+ and T− correspond to hopping processes that

increase and decrease energies by one unit with respect to the CoM term C. That

is,

C |µ〉 = Eµ |µ〉 =⇒ C(T± |µ〉

)= (Eµ ± 1) T± |µ〉 . (E.4)

Following the standard Schrieffer-Wolff procedure [246], we divide the Hilbert

space into “blocks”, which are subspaces degenerate under the leading order term

C. Terms of the Hamiltonian which only have non-vanishing matrix elements

within the same block are called “block diagonal” whereas terms which only

have non-vanishing matrix elements between different blocks are called “block off-

diagonal”. For the Hamiltonian H, the “block diagonal” and “block off-diagonal”

parts Hd and Hod respectively read

H = C + λV︸ ︷︷ ︸Hd

+ λT︸︷︷︸Hod

. (E.5)

Next, we wish to perturbatively find a unitary transformation such that the

277

resultant Hamiltonian has no “block off-diagonal” parts:

Heff = eλSHe−λS =∞∑n=0

λnH(n)eff , (E.6)

where S is anti-Hermitian, and H(n)eff is the effective Hamiltonian in n-th order

perturbation theory Here, each block diagonal subspace of Heff corresponds to a

subspace degenerate under C—since C is the center-of-mass operator with OBC

(see Eq. (5.3)), different block diagonal parts of Heff correspond to subspaces la-

belled by distinct center-of-mass quantum numbers. In what follows, we show that

the pair-hopping term arises in H(3)eff i.e., in the effective Hamiltonian restricted to

one center-of-mass sector of the Bloch MBL Hamiltonian.

We now derive the expression for Heff up to third order in perturbation theory.

Expanding Heff, defined in Eq. (E.6), in powers of λ, we obtain

Heff = H + λ [S,H] +λ2

2[S, [S,H]] +

λ3

6[S, [S, [S,H]]] +O

(λ4). (E.7)

We also expand S in powers of λ as

S = S0 + λS1 + λ2S2 +O(λ3). (E.8)

Using Eqs. (E.5) and (E.7), we obtain

Heff = C + λV + T +

[S0, C

]+ λ2

12

[S0,[S0, C

]]+[S0, V + T

]+[S1, C

]+λ3

16

[S0,[S0,[S0, C

]]]+ 1

2

([S1,[S0, C

]]+[S0,[S1, C

]]+[S0,[S0, V + T

]])+[S1, V + T

]+[S2, C

]+O (λ4) . (E.9)

278

Since V is diagonal, to cancel the block off-diagonal component T at O (λ) in

Eq. (E.9) we require that S0 satisfies

[S0, C

]= −T . (E.10)

Simplifying Eq. (E.9) using Eq. (E.10), we obtain

Heff = C + λV + λ2

12

[S0, T

]+[S0, V

]+[S1, C

]+λ3

13

[S0,[S0, T

]]+ 1

2

([S1, T

]+[S0,[S1, C

]]+[S0,[S0, V

]])+[S1, V

]+[S2, C

]+O (λ4) . (E.11)

To determine the block off-diagonal terms at O (λ2) in Eq. (E.11), we note that

since C and T are block diagonal and block off-diagonal respectively, we can

always choose S0 in Eq. (E.10) to be block off-diagonal. Thus,[S0, T

]can have

block diagonal terms, whereas[S0, V

]is completely block off-diagonal. To cancel

the block off-diagonal terms at O (λ2) in Eq. (E.11), we hence require that S1

satisfies [S1, C

]= −

[S0, V

]− 1

2

([S0, T

]− P

[S0, T

]P), (E.12)

where P is a projector that kills block off-diagonal components. That is, if a

matrix X has both block diagonal and block off-diagonal components, PXP (resp.

(X − PXP)) is completely block diagonal (resp. off-diagonal). Simplifying the

expression for Heff in Eq. (E.11) using Eq. (E.12), we obtain

Heff = C+λV+λ2

2P[S0, T

]P+λ3

1

3

[S0,[S0, T

]]+

1

2

[S1, T

]+[S1, V

]+[S2, C

]+O

(λ4).

(E.13)

279

In Eq. (E.12), since the RHS is block off-diagonal, and C is block diagonal, S1

can be chosen to be block off-diagonal. Since S0 and S1 are block off-diagonal,[S1, T

]and

[S0,[S0, T

]]can have block diagonal components whereas

[S1, V

]is completely block off-diagonal. Moreover, in the Schrieffer-Wolff procedure, S2

is chosen such that the term[S2, C

]cancels block off-diagonal terms at O (λ3).

That is,

[S2, C

]= −

[S1, V

]− 1

2

([S1, T

]− P

[S1, T

]P)

−13

([S0,[S0, T

]]− P

[S0,[S0, T

]]P). (E.14)

Thus, Heff reads

Heff = C +λV +λ2

2P[S0, T

]P +λ3P

(1

2

[S1, T

]+

1

3

[S0,[S0, T

]])P +O

(λ4).

(E.15)

Thus, we find that H(2)eff and H

(3)eff are given by

H(2)eff =

1

2P[S0, T

]P , H

(3)eff = P

(1

2

[S1, T

]+

1

3

[S0,[S0, T

]])P . (E.16)

We now compute S0 and S1 in order to obtain the effective Hamiltonians H(2)eff and

H(3)eff . According to Eq. (E.10), S0 is determined by

[S0, C

]= −T = −

(T+ + T−

). (E.17)

We first compute some useful commutators:

[T+, T−

]= 0,

[T+, C

]= −T+,

[T−, C

]= T−, (E.18)

280

Thus, Eq. (E.10) is satisfied by choosing S0 = T+ − T−. Note that this choice of

S0 is block off-diagonal and anti-Hermitian. Thus, using Eq. (E.18), we obtain[S0, T

]= 0 and H

(2)eff = 0. S1 is computed using Eq. (E.12), and the relevant

commutators read

[T+, V1

]= α1

∑j

(nj−1c

†j+1cj − c

†jcj−1nj+1

)≡ α1

(O+− − O++

)[T−, V1

]= α1

∑j

(−nj−1c

†jcj+1 + c†j−1cjnj+1

)≡ α1

(O−+ − O−−

),

[T+, V0

]= α0

∑j

(wj − wj+1) c†j+1cj ≡ α0

(−F+ + B+

)[T−, V0

]= α0

∑j

(wj+1 − wj) c†jcj+1 ≡ α0

(F− − B−

)=⇒

[S0, V

]= α1

(O+− + O−− − O−+ − O++

)+ α0

(−F+ − F− + B+ + B−

).

(E.19)

where we have defined the operators

O++ =∑j

c†jcj−1nj+1, O−+ =∑j

c†j−1cjnj+1, O+− =∑j

c†j+1cjnj−1, O−− =∑j

c†jcj+1nj−1,

F+ =∑j

wj+1c†j+1cj, F− =

∑j

wj+1c†jcj+1, B+ =

∑j

wjc†j+1cj, B− =

∑j

wjc†jcj+1. (E.20)

Thus, according to Eq. (E.12), S1 should satisfy

[S1, C

]= −

[S0, V

]= α1

(−O+− − O−− + O−+ + O++

)+ α0

(F+ + F− − B+ − B−

).

(E.21)

We now show that S1 can be chosen to be a linear superposition of Oµν , Fµ, and

281

Bµ, where µ, ν ∈ +,−. The commutators[Oµν , C

],[Fµ, C

], and

[Bµ, C

]read

[O++, C

]= −O++,

[O−+, C

]= O−+,

[O+−, C

]= −O+−,

[O−−, C

]= O−−,[

F+, C]= −F+,

[F−, C

]= F−,

[B+, C

]= −B+,

[B−, C

]= B−. (E.22)

Thus, Eq. (E.21) is satisfied by choosing

S1 = α1

(O+− − O−− + O−+ − O++

)+ α0

(−F+ + F− + B+ − B−

). (E.23)

Noting that[S0, T

]= 0, H(3)

eff in Eq. (E.16) reads

H(3)eff =

λ3

2P[S1, T+ + T−

]P = −α1λ

3

2P[T+ + T−, O+− − O−− + O−+ − O++

]P

−α0λ3

2P[T+ + T−,−F+ + F− + B+ − B−

]P .

(E.24)

Note that

P[T+, O+−

]P = P

[T−, O−−

]P = 0, P

[T+, O++

]P = P

[T−, O−+

]P = 0

P[T+, F+

]P = P

[T−, F−

]P = 0, P

[T+, B+

]P = P

[T−, B−

]P = 0, (E.25)

since these terms change the energy of eigenstates of C, and are hence block

282

off-diagonal. Simplifying Eq. (E.24) using Eq. (E.25), we obtain

H(3)eff = −λ3

2Pα1

(−[T+, O−−

]+[T+, O−+

])+ α0

([T+, F−

]−[T+, B−

])+ h.c.

P

= −λ3∑j

[α1

(c†jc

†j+3cj+2cj+1 + h.c.

)+ 2 (njnj+1 − njnj+2)

− α0(2wj − wj−1 − wj+1) nj

](E.26)

Finally, re-introducing the overall factor of E, the full effective Hamiltonian re-

stricted to one center-of-mass sector is:

Heff = V0∑j

wjnj + V1∑j

njnj+1 + V2∑j

njnj+2 − t2V1E2

∑j

(c†jc

†j+3cj+2cj+1 + h.c.

)+O

(t3

E3

),

(E.27)

where we have omitted the term E∑j

jnj since it is a symmetry of the effective

Hamiltonian, and we have defined

wj ≡(1− 2t2

E2

)wj +

t2

E2(wj−1 + wj+1) , V1 ≡ V1

(1− 2t2

E2

), V2 ≡

2t2V1E2

.

(E.28)

E.1.2 Dynamical connections to Bloch MBL

As evident from Eq. (E.27), the leading order hopping term in the effective Hamil-

tonian governing the Wannier-Stark model is the pair-hopping term studied in

this Chapters 5 and 6, given by Eq. (5.1). Longer range center-of-mass preserving

terms, including n-body terms for n > 2 appear at higher orders in perturbation

theory, and are therefore suppressed by higher powers of t/E; we thus expect

their strength to drop off exponentially with range as ∼ tn/En, for terms which

283

have support over ∼ n sites. Given this mapping, we now comment briefly on the

phenomenon of Bloch MBL, as discussed in Refs. [223, 224]. We begin by noting

that the electric field in itself is not sufficient to give MBL, since while the electric

field ‘switches off’ single particle hopping, it leaves in place the correlated center-

of-mass preserving hopping processes discussed above. As we have discussed in

the preceding sections, eigenstates of such processes are by no means guaranteed

to be localized. Thus, different physics must underlie the numerical observation

of MBL in the Bloch MBL problem.

Strictly in the E/t → ∞ limit, the effective Hamiltonian consists only of the

nearest-neighbor electrostatic term V1∑

j njnj+1 and the onsite potential term

V0∑

j wjnj. When wj = 0, i.e. without disorder or curvature, the eigenstates

are clearly not localized since the spectrum of V1∑

j njnj+1 is highly degenerate.

However, that degeneracy is lifted by small disorder or curvature; thus, when wj

is random or wj ∼ j2, all the eigenstates of Eq. (E.27) have low entanglement.

This is consistent with the fact that Refs. [223,224] do not observe MBL without

curvature or disorder respectively.

Moving away from the E/t→∞ limit, we obtain the effective Hamiltonian of

Eq. (E.27) for large but finite E/t, which exhibits Krylov fracture. The fracture

is said to be ‘strong’ [86, 87] if the dimension of the largest Krylov subspace is

a vanishing fraction of the full Hilbert space dimension in the thermodynamic

limit. This leads to the non-thermalization of generic initial product states with

respect to the entire Hilbert space [86,87], for example the entanglement entropy

does not saturate to the maximum value allowed by the full Hilbert space. For a

‘minimal’ center-of-mass preserving Hamiltonian, such as the pair-hopping model

of Eq. (5.1), obtained by retaining only the leading order hopping terms in the

284

0 10 20 30 40 50

E/t

0.0

0.2

0.4

0.6

0.8

1.0

⟨ O⟩ T

L = 6L = 8L = 10L = 12L = 14

0 10 20 30 40 50

T

0.0

0.2

0.4

0.6

0.8

1.0

O E/t = 4

E/t = 20

Figure E.1: Inset: Weight of the state e−iHBlochT |ψ0〉 within the Krylov subspaceK (Heff, |ψ0〉), captured by the quantityO(T ) (defined in Eq. (E.29)) for two valuesof the electric field E. Main: Time-average of O (T ), denoted by 〈O〉T as afunction of E/t. O is close to 1 for larger values of E, justifying that the pair-hopping Hamiltonian H is a good approximation for HBloch. Data is shown forV0/t = 0, V1/t = 1, and |ψ0〉 = |↑↓↑↓ · · · · · · 〉 = |01100110 · · · · · · 〉.

effective Hamiltonian, strong fracture indeed occurs.∗ A simple example of such

non-thermalization is,the CDW state |0101 · · · 01〉 used as a diagnostic of localiza-

tion in Ref. [224]. This state forms a one-dimensional Krylov subspace under the

pair-hopping Hamiltonian of Eq. (5.1): it maps onto the state |↑↑ . . . ↑↑〉 under

the mapping defined in Sec. 5.3. Clearly, once initialized with this state, the sys-

tem will forever retain memory of its initial condition under time evolution with

the minimal pair-hopping Hamiltonian.

However, we note that the effective Hamiltonian Heff of Eq. (E.27) is a good

approximation to the Bloch MBL Hamiltonian HBloch, given by Eq. (E.1), only∗The pair-hopping Hamiltonian Eq. (5.1) is equivalent to a S = 1/2 spin Hamiltonian for

which evidence of strong fracture was found in Ref. [86]. We have also verified numerically upto L = 24 that the size of the largest Krylov subspace ∼ 2L while the Hilbert space dimension∼ 4L, consistent with strong fracture.

285

for large values of E/t. To test the effectiveness of Heff, we study the quantity

O (T ) =∑

|ϕn⟩∈K(Heff,|ψ0⟩)

| 〈ϕn| e−iHBlochT |ψ0〉 |2, (E.29)

which is the weight of the state e−iHBlochT |ψ0〉 within the Krylov subspaceK (Heff, |ψ0〉).†

We expect Heff to correctly capture the dynamics of HBloch only for values of E/t

when

〈O〉T ≡ limτ→∞

1

τ

∫dτ O (τ) ≈ 1. (E.30)

In Fig. E.1, we show the behavior of 〈O〉T for the initial state |ψ0〉 = |↑↓↑↓ · · · · · · 〉 =

|01100110 · · · · · · 〉. Thus, we find that Heff is a good approximation for HBloch only

for E/t ≳ 50 when V0, V1 ∼ O (1) and for system sizes up to L = 14. In Fig. E.1,

we also find that for a fixed value of E/t, Heff becomes a worse approximation for

HBloch with increasing system size. Thus, it is not clear whether Krylov fracture

of the pair-hopping model of Eq. (5.1) plays a significant role in the observations

of Refs. [223, 224], which focus on the regimes where E/t ∼ O (10).

To conclude this section, we speculate on two mechanisms that give rise to lo-

calized eigenstates at the smaller values of E/t with disorder, which could provide

a partial explanation for the Bloch MBL phenomenon in Refs. [223, 224]: (i) At

smaller values of E/t, terms at higher order in perturbation theory cannot be

neglected in the effective Hamiltonian. However, since terms generated at all or-

ders in perturbation theory are necessarily center-of-mass preserving, the hopping

term of the effective Hamiltonian at any finite order exhibits exponentially many

frozen eigenstates [86,87]. The addition of disorder breaks the exponentially large†Note that since Heff of Eq. (E.27) and H of Eq. (5.1) only differ by diagonal terms,

K (Heff, |ψ0〉) = K (H, |ψ0〉).

286

degeneracy of these frozen states under the effective Hamiltonian, which results

in exponentially many product eigenstates of the effective Hamiltonian at any

finite order. (ii) When disorder is added in Eq. (E.1) (as is done in Ref. [223]),

then this can give rise to conventional ‘disorder-induced’ MBL [38] within Krylov

subspaces of the effective Hamiltonian. This can happen even when the disorder

is weak compared to the bare single particle hopping t, because the disorder may

be strong compared to the largest hopping term: from Eq. (E.27), we see that

the hopping term is of O (t2V1/E2), while the disorder is an O (V0) term, which

suggests the possibility of conventional MBL in the effective Hamiltonian.

E.2 Quantum Hall Effect

E.2.1 One-dimensional Mapping of a Landau level

We briefly review the origin of 1D pair-hopping models via a mapping of a sin-

gle Landau level of a 2D quantum Hall system on a cylinder/torus to a 1D

chain [215, 216, 229, 247, 248]. For the sake of illustration, we consider electrons

either on an infinite cylinder for open boundary conditions (OBC) or on a torus

for periodic boundary conditions (PBC) of length Lx and circumference Ly with

NΦ = LxLy/(2π) flux quanta. We consider the Landau gauge A = Bxy, where B

is the transverse magnetic field and y is the direction along the circumference of

the cylinder. Setting the magnetic length to√

~eB

= 1, within a Landau level, we

define the single-particle magnetic translation operators as [216, 249]

tx = exp

(LxNΦ

(∂

∂x+ iy

)), ty = exp

(LyNΦ

∂

∂y

), (E.31)

287

which obey the commutation relation [247]

txty = exp

(−i 2πNΦ

)ty tx. (E.32)

A complete orthornormal basis of single-particle orbitals ψl,j(r), 1 ≤ j ≤ NΦ in

the l-th Landau level can be constructed using eigenstates of ty. These eigenstates

satisfy

tyψl,j (r) = exp

(i2πj

NΦ

)ψl,j (r) , txψl,j (r) = ψl,j+1 (r) . (E.33)

For example, these wavefunctions in the lowest Landau level (l = 0) on a torus

read [216, 248]

ψ0,j (r) =1√√πLy

∞∑m=−∞

[exp

(iyLx

(m+

j

NΦ

))exp

(−1

2

(x+

(m+

j

NΦ

)Lx

)2)]

.

(E.34)

For spinless electrons within a Landau level, the electron-electron interaction term

can be written in the second-quantized form [216]

Hl =

NΦ−1∑j1,j2,j3,j4=0

V(l)j1,j2,j3,j4

c†j1c†j2cj3cj4 , (E.35)

where c†j and cj are the fermionic creation and annihilation operators for the

single-particle orbital ψl,j(r), and

V(l)j1,j2,j3,j4

≡ 1

2

∫∫T2

d2r1 d2r2(ψ∗l,j1

(r1)ψ∗l,j2

(r2)V (r1 − r2)ψl,j3 (r2)ψl,j4 (r1)).

(E.36)

288

Since j1, j2, j3 and j4 are the y momentum eigenvalues (see Eq. (E.33)), for any

interaction V that is translation invariant in the y direction we obtain

j1 + j2 = j3 + j4 mod NΦ. (E.37)

Further using translation invariance in the x direction, the Hamiltonian Hl of

Eq. (E.35) can be reparametrized as [215, 216, 229]

Hl =

NΦ2

−1∑m=0

NΦ2∑

k=m+1

[V

(l)k,m

(1+δm,0)(1+δk,NΦ/2)

NΦ−1∑j=0

(c†jc

†j+k+mcj+kcj+m + h.c.

)]

≡NΦ2

−1∑m=0

NΦ2∑

k=m+1

V(l)k,mCk,m, (E.38)

where

V(l)k,m ≡ V

(l)j+m,j+k,j+k+m,j − V

(l)j+m,j+k,j,j+k+m + V

(l)j+k,j+m,j,j+k+m − V

(l)j+k,j+m,j+k+m,j.

(E.39)

A procedure for obtaining V (l)k,m given the potential V (r) is outlined in App. E.2.2.

We also refer readers to Ref. [217] for a more general analysis. For example,

consider the short-range Haldane-Trugman-Kivelson potential, [250, 251]

V (r1 − r2) ∝ ∇2δ (r1 − r2) . (E.40)

The matrix elements V (0)k,m for this potential in the lowest Landau level follow

(when Lx, NΦ →∞) (see Eq. (E.57))

V(0)k,m ∝

(k2 −m2

)e−2π2 k2+m2

L2y . (E.41)

289

Note that whenever m = 0, Ck,0 of Eq. (E.38) reads

Ck,0 =

NΦ−1∑j=0

c†jcjc†j+kcj+k ≡

NΦ−1∑j=0

njnj+k, (E.42)

where nj ≡ c†jcj. Thus Ck,0 is a pure electrostatic term. In the thin-torus limit

(Ly → 0), the strength of the terms Ck,m decreases exponentially with increasing

(k2+m2) (see Eqs. (E.41) and (E.57)). Thus, taking into account the terms up to

the largest non-electrostatic term for a short-range potential in the lowest Landau

level (remembering k ≥ m+ 1), we obtain an effective Hamiltonian HLLL

HLLL =NΦ−1∑j=0

(V

(0)1,0 njnj+1 + V

(0)2,0 njnj+2 + V

(0)2,1

(c†jc

†j+3cj+2cj+1 + h.c.

)).(E.43)

We refer to the term C2,1 as the “pair-hopping term”. Note that these Hamiltoni-

ans can be generalized to three-body hopping terms, but we do not consider them

in this work.

E.2.2 Obtaining V(l)k,mor general potentials

In this appendix, we outline a general procedure for obtaining V(l)k,m for general

potentials in the l-th Landau level on the cylinder geometry. Note that similar

calculations have been performed in the literature, for example in Refs. [217]

and [252]. We start with the Fourier representation of the potential

V (r) =∑q

V (q) eiq·r. (E.44)

290

If ψl,α (r) are the single-particle orbitals in the l-th Landau level, we obtain

V(l)j1,j2,j3,j4

using Eq. (E.36):

V(l)j1,j2,j3,j4

≡ 1

2

∫∫d2r1 d2r2 ψ

∗l,j1

(r1)ψ∗l,j2

(r2)V (r1 − r2)ψl,j3 (r2)ψl,j4 (r1)

≡ 1

2

∑q

V (q) I(l)j1,j4

(q) I(l)j2,j3

(−q), (E.45)

where we have defined

I(l)α,β (q) ≡

∫d2r ψ∗

l,α (r) eiq·rψl,β (r). (E.46)

Using Eq. (E.39), the expression for V (l)k,m can then be written as

V(l)k,m = 1

2

∑q

V (q)[I(l)j+m,j (q) I

(l)j+k,j+k+m (−q)− I(l)j+m,j+k+m (q) I

(l)j+k,j (−q)

+I(l)j+k,j+k+m (q) I

(l)j+m,j (−q)− I

(l)j+k,j (q) I

(l)j+m,j+k+m (−q)

]≡ 1

2

∑q

V (q) J(l)k,m (q).(E.47)

We now compute J(l)k,m (q), which is independent of the potential V (q). As in

the main text, we work in the Landau gauge where A = Byx. On the cylinder

geometry, the single-particle wavefunctions read

ψl,α (r) =1√√π2ll!Ly

Hl

(x+

2πα

Ly

)e− 1

2

(x+ 2πα

Ly

)2

e2πiαyLy , (E.48)

where Hl(x) is the l-th Hermite polynomial in x. Using Eqs. (E.48) and (E.46),

we obtain the expression for I(l)α,β (q) as

I(l)α,β (q) = δ 2π(β−α)

Ly,qye− |q|2

4−πi(α+β)qx

Ly Ll

(|q|2

2

), (E.49)

291

where we have used the identity [253]

1

2ll!√π

∫ ∞

−∞dx Hl (x+ ia)Hl (x+ ib) e−x

2

= Ll (2ab) , (E.50)

where Ll(x) is the l-th Laguerre polynomial in x. Before computing J (l)k,m (q), for

convenience we compute

K(l)α,β,γ (q) ≡ I

(l)α,α+γ (q) I

(l)β+γ,β (−q) = e−

|q|22

(Ll

(|q|2

2

))2

δ 2πγLy

,qye

2πi(β−α)qxLy .

(E.51)

Using Eqs. (E.47) and (E.51), J (l)km (q) can then be written as

J(l)km(q) = K

(l)j+m,j+k+m,−m(q)−K

(l)j+m,j,k(q) +K

(l)j+k,j,m (q)−K(l)

j+k,j+k+m,−k (q)

= e−|q|22

(Ll

(|q|22

))2 [δ− 2πm

Ly,qye

2πikqxLy − δ 2πk

Ly,qye− 2πimqx

Ly + δ 2πmLy

,qye− 2πikqx

Ly − δ− 2πkLy

,qye

2πimqxLy

].

(E.52)

V(l)k,m can then be computed using Eq. (E.47). In fact, it is convenient to write

V(l)k,m in terms of W (l)

k,m, which is defined as

W(l)k,m =

∑q

V (q)e−|q|22

(Ll

(|q|22

))2δ 2πm

Ly,qye

2πikqxLy ,

Vk,m = W(l)k,−m −W

(l)−m,k +W

(l)−k,m −W

(l)m,−k. (E.53)

W(l)k,m can be simplified to

W(l)k,m = e−

A2+B2

2

∫ ∞

−∞dx V (x,B)

(Ll

(x2 +B2

2

))2

e−(x−iA)2

2 ,

(E.54)

292

where we have defined A = 2πkLy

, B = 2πmLy

.

We now illustrate an example using the short-range Haldane-Trugman-Kivelson

potential, [250, 251]

V (r) = ∇2δ(2)(r) = −∑q

|q|2eiq·r =⇒ V (qx, qy) = −(q2x + q2y

). (E.55)

We compute W (l)k,m using Eq. (E.54), which reduces to

W(l)k,m = −e

−2π2(k2+m2)

L2y

∫ ∞

−∞dx (x2 +B2)

(Ll

(x2 +B2

2

))2

e−(x−iA)2

2 . (E.56)

We have not been able to obtain a useful closed form expression for W (l)k,m of

Eq. (E.56) for general l. Here we list V (l)k,m for the lowest two Landau levels:

V(0)k,m = 16π2

L2y(k −m)(k +m)e

−2π2(k2+m2)

L2y

V(1)k,m = 16π2

L2y(k −m)(k +m)

(15− 24π2

L2y(k2 +m2) + 16π4

L4y(k2 −m2)

2)e−

2π2(k2+m2)L2y .(E.57)

293

References

[1] T. Kinoshita, T. Wenger, and D. S. Weiss. A quantum Newton’s cradle.Nature, 440:900–903, April 2006.

[2] M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl,I. Mazets, D. Adu Smith, E. Demler, and J. Schmiedmayer. Relaxation andPrethermalization in an Isolated Quantum System. Science, 337(6100):1318,Sep 2012.

[3] Michael Schreiber, Sean S. Hodgman, Pranjal Bordia, Henrik P. Lüschen,Mark H. Fischer, Ronen Vosk, Ehud Altman, Ulrich Schneider, andImmanuel Bloch. Observation of many-body localization of interactingfermions in a quasirandom optical lattice. Science, 349(6250):842–845, Aug2015.

[4] J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess, P. Hauke, M. Heyl,D. A. Huse, and C. Monroe. Many-body localization in a quantum simulatorwith programmable random disorder. Nature Physics, 12(10):907–911, Oct2016.

[5] Adam M. Kaufman, M. Eric Tai, Alexander Lukin, Matthew Rispoli, RobertSchittko, Philipp M. Preiss, and Markus Greiner. Quantum thermaliza-tion through entanglement in an isolated many-body system. Science,353(6301):794–800, Aug 2016.

[6] Hannes Bernien, Sylvain Schwartz, Alexander Keesling, Harry Levine,Ahmed Omran, Hannes Pichler, Soonwon Choi, Alexander S. Zibrov,Manuel Endres, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin.Probing many-body dynamics on a 51-atom quantum simulator. Nature,551(7682):579–584, Nov 2017.

[7] G. Kucsko, S. Choi, J. Choi, P. C. Maurer, H. Zhou, R. Landig, H. Sumiya,S. Onoda, J. Isoya, F. Jelezko, E. Demler, N. Y. Yao, and M. D. Lukin. Crit-ical thermalization of a disordered dipolar spin system in diamond. PhysicalReview Letters, 121:023601, Jul 2018.

294

[8] Subir Sachdev and Jinwu Ye. Gapless spin-fluid ground state in a randomquantum heisenberg magnet. Physical Review Letters, 70:3339–3342, May1993.

[9] Alexei Kitaev. A simple model of quantum holography. In KITP stringsseminar and Entanglement, volume 12, 2015.

[10] Juan Maldacena and Douglas Stanford. Remarks on the sachdev-ye-kitaevmodel. Physical Review D, 94:106002, Nov 2016.

[11] Joseph Polchinski and Vladimir Rosenhaus. The spectrum in the Sachdev-Ye-Kitaev model. Journal of High Energy Physics, 2016:1, 2016.

[12] Jordan S. Cotler, Guy Gur-Ari, Masanori Hanada, Joseph Polchinski, PhilSaad, Stephen H. Shenker, Douglas Stanford, Alexandre Streicher, andMasaki Tezuka. Black holes and random matrices. Journal of High En-ergy Physics, 2017:118, 2017.

[13] Jordan Cotler, Nicholas Hunter-Jones, Junyu Liu, and Beni Yoshida. Chaos,complexity, and random matrices. Journal of High Energy Physics, 2017:48,2017.

[14] J. M. Deutsch. Quantum statistical mechanics in a closed system. PhysicalReview A, 43:2046–2049, Feb 1991.

[15] Mark Srednicki. Chaos and quantum thermalization. Physical Review E,50:888–901, Aug 1994.

[16] Marcos Rigol, Vanja Dunjko, and Maxim Olshanii. Thermalization and itsmechanism for generic isolated quantum systems. Nature, 452(7189):854–858, Apr 2008.

[17] Anatoli Polkovnikov, Krishnendu Sengupta, Alessandro Silva, and MukundVengalattore. Colloquium: Nonequilibrium dynamics of closed interactingquantum systems. Reviews of Modern Physics, 83:863–883, Aug 2011.

[18] Luca D’Alessio, Yariv Kafri, Anatoli Polkovnikov, and Marcos Rigol. Fromquantum chaos and eigenstate thermalization to statistical mechanics andthermodynamics. Advances in Physics, 65(3):239–362, May 2016.

[19] Christian Gogolin and Jens Eisert. Equilibration, thermalisation, and theemergence of statistical mechanics in closed quantum systems. Reports onProgress in Physics, 79(5):056001, May 2016.

295

[20] Takashi Mori, Tatsuhiko N. Ikeda, Eriko Kaminishi, and Masahito Ueda.Thermalization and prethermalization in isolated quantum systems: atheoretical overview. Journal of Physics B Atomic Molecular Physics,51(11):112001, Jun 2018.

[21] David J Luitz, Nicolas Laflorencie, and Fabien Alet. Many-body localizationedge in the random-field heisenberg chain. Physical Review B, 91(8):081103,2015.

[22] L. D. Faddeev. How algebraic bethe ansatz works for integrable model.arXiv: hep-th/9605187, 1996.

[23] Marcos Rigol. Breakdown of thermalization in finite one-dimensional sys-tems. Physical Review Letters, 103(10):100403, 2009.

[24] Christian Gogolin, Markus P Müller, and Jens Eisert. Absence of thermal-ization in nonintegrable systems. Physical Review Letters, 106(4):040401,2011.

[25] Balázs Pozsgay. Failure of the generalized eigenstate thermalization hypoth-esis in integrable models with multiple particle species. Journal of StatisticalMechanics: Theory and Experiment, (9):P09026, 2014.

[26] P. W. Anderson. Absence of diffusion in certain random lattices. PhysicalReview, 109:1492–1505, Mar 1958.

[27] D. M. Basko, I. L. Aleiner, and B. L. Altshuler. Metal insulator transitionin a weakly interacting many-electron system with localized single-particlestates. Annals of Physics, 321(5):1126–1205, May 2006.

[28] Arijeet Pal and David A Huse. Many-body localization phase transition.Physical Review B, 82(17):174411, 2010.

[29] John Z Imbrie. On many-body localization for quantum spin chains. Journalof Statistical Physics, 163(5):998–1048, 2016.

[30] Michael Gring, Maximilian Kuhnert, Tim Langen, Takuya Kitagawa, Bern-hard Rauer, Matthias Schreitl, Igor Mazets, D Adu Smith, Eugene Demler,and Jörg Schmiedmayer. Relaxation and prethermalization in an isolatedquantum system. Science, 337(6100):1318–1322, 2012.

[31] Jack Kemp, Norman Y Yao, Christopher R Laumann, and Paul Fendley.Long coherence times for edge spins. Journal of Statistical Mechanics :Theory and Experiment, page 063105, 2017.

296

[32] Dominic V Else, Paul Fendley, Jack Kemp, and Chetan Nayak. Prethermalstrong zero modes and topological qubits. Physical Review X, 7(4):041062,2017.

[33] David A Huse, Rahul Nandkishore, Vadim Oganesyan, Arijeet Pal, andSL Sondhi. Localization-protected quantum order. Physical Review B,88(1):014206, 2013.

[34] Anushya Chandran, Vedika Khemani, CR Laumann, and SL Sondhi. Many-body localization and symmetry-protected topological order. Physical Re-view B, 89(14):144201, 2014.

[35] Yasaman Bahri, Ronen Vosk, Ehud Altman, and Ashvin Vishwanath. Lo-calization and topology protected quantum coherence at the edge of hotmatter. Nature Communications, 6, 2015.

[36] Vedika Khemani, Roderich Moessner, and S. L. Sondhi. A brief history oftime crystals. arXiv: 1910.10745, 2019.

[37] Vedika Khemani, DN Sheng, and David A Huse. Two universalityclasses for the many-body localization transition. Physical Review Letters,119(7):075702, 2017.

[38] Rahul Nandkishore and David A. Huse. Many-Body Localization and Ther-malization in Quantum Statistical Mechanics. Annual Review of CondensedMatter Physics, 6:15–38, Mar 2015.

[39] Ehud Altman and Ronen Vosk. Universal Dynamics and Renormalizationin Many-Body-Localized Systems. Annual Review of Condensed MatterPhysics, 6:383–409, Mar 2015.

[40] Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn.Colloquium: Many-body localization, thermalization, and entanglement.Reviews of Modern Physics, 91:021001, May 2019.

[41] Ronen Vosk and Ehud Altman. Many-body localization in one dimensionas a dynamical renormalization group fixed point. Physical Review Letters,110(6):067204, 2013.

[42] David A. Huse, Rahul Nandkishore, and Vadim Oganesyan. Phenomenologyof fully many-body-localized systems. Physical Review B, 90:174202, Nov2014.

[43] Wojciech De Roeck and François Huveneers. Scenario for delocalization intranslation-invariant systems. Physical Review B, 90:165137, Oct 2014.

297

[44] Wojciech De Roeck and François Huveneers. Asymptotic Quantum Many-Body Localization from Thermal Disorder. Communications in Mathemat-ical Physics, 332(3):1017–1082, Dec 2014.

[45] Tarun Grover and Matthew P A Fisher. Quantum disentangled liquids.Journal of Statistical Mechanics: Theory and Experiment, 2014(10):P10010,oct 2014.

[46] Mauro Schiulaz, Alessandro Silva, and Markus Müller. Dynamics in many-body localized quantum systems without disorder. Physical Review B,91:184202, May 2015.

[47] Z Papić, E. M. Stoudenmire, and D. A.” Abanin. Many-body localizationin disorder-free systems: The importance of finite-size constraints. Annalsof Physics, 362:714 – 725, 2015.

[48] N. Y. Yao, C. R. Laumann, J. I. Cirac, M. D. Lukin, and J. E. Moore. Quasi-many-body localization in translation-invariant systems. Physical ReviewLetters, 117:240601, Dec 2016.

[49] A. Smith, J. Knolle, D. L. Kovrizhin, and R. Moessner. Disorder-free local-ization. Physical Review Letters, 118:266601, Jun 2017.

[50] A. Smith, J. Knolle, R. Moessner, and D. L. Kovrizhin. Absence of ergodicitywithout quenched disorder: From quantum disentangled liquids to many-body localization. Physical Review Letters, 119:176601, Oct 2017.

[51] Alexios A. Michailidis, Marko Znidaric, Mariya Medvedyeva, Dmitry A.Abanin, Tomaz Prosen, and Z. Papic. Slow dynamics in translation-invariant quantum lattice models. Physical Review B, 97:104307, Mar 2018.

[52] Marlon Brenes, Marcello Dalmonte, Markus Heyl, and Antonello Scardic-chio. Many-body localization dynamics from gauge invariance. PhysicalReview Letters, 120:030601, Jan 2018.

[53] Naoto Shiraishi and Takashi Mori. Systematic construction of counterex-amples to eigenstate thermalization hypothesis. Physical Review Letters,119:030601, 2017.

[54] Ian Affleck, Tom Kennedy, Elliott H. Lieb, and Hal Tasaki. Rigorous resultson valence-bond ground states in antiferromagnets. Physical Review Letters,59(7):799–802, Aug 1987.

298

[55] Sanjay Moudgalya, Stephan Rachel, B. Andrei Bernevig, and Nicolas Reg-nault. Exact excited states of nonintegrable models. Physical Review B,98:235155, Dec 2018.

[56] Sanjay Moudgalya, Nicolas Regnault, and B. Andrei Bernevig. Entangle-ment of exact excited states of Affleck-Kennedy-Lieb-Tasaki models: Exactresults, many-body scars, and violation of the strong eigenstate thermaliza-tion hypothesis. Physical Review B, 98:235156, Dec 2018.

[57] Michael Schecter and Thomas Iadecola. Weak ergodicity breaking andquantum many-body scars in spin-1 xy magnets. Physical Review Letters,123:147201, Oct 2019.

[58] Thomas Iadecola and Michael Schecter. Quantum many-body scar stateswith emergent kinetic constraints and finite-entanglement revivals. PhysicalReview B, 101:024306, Jan 2020.

[59] Sambuddha Chattopadhyay, Hannes Pichler, Mikhail D. Lukin, andWen Wei Ho. Quantum many-body scars from virtual entangled pairs.Physical Review B, 101:174308, May 2020.

[60] Naoyuki Shibata, Nobuyuki Yoshioka, and Hosho Katsura. Onsager’s scarsin disordered spin chains. Phys. Rev. Lett., 124:180604, May 2020.

[61] Daniel K. Mark, Cheng-Ju Lin, and Olexei I. Motrunich. Unified structurefor exact towers of scar states in the affleck-kennedy-lieb-tasaki and othermodels. Physical Review B, 101:195131, May 2020.

[62] Kieran Bull, Jean-Yves Desaules, and Zlatko Papić. Quantum scars as em-beddings of weakly broken lie algebra representations. Physical Review B,101:165139, Apr 2020.

[63] Sanjay Moudgalya, Edward O’Brien, B. Andrei Bernevig, Paul Fendley,and Nicolas Regnault. Large classes of quantum scarred hamiltonians frommatrix product states. arXiv: 2002.11725, 2020.

[64] Sanjay Moudgalya, Nicolas Regnault, and B. Andrei Bernevig. Eta-pairingin hubbard models: From spectrum generating algebras to quantum many-body scars. arXiv: 2004.13727, 2020.

[65] Daniel K. Mark and Olexei I. Motrunich. Eta-pairing states as true scars inan extended hubbard model. arXiv: 2004.13800, 2020.

299

[66] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić.Weak ergodicity breaking from quantum many-body scars. Nature Physics,14(7):745–749, 2018.

[67] Michael Schecter and Thomas Iadecola. Many-body spectral reflection sym-metry and protected infinite-temperature degeneracy. Physical Review B,98:035139, Jul 2018.

[68] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Pa-pić. Quantum scarred eigenstates in a rydberg atom chain: Entanglement,breakdown of thermalization, and stability to perturbations. Physical Re-view B, 98:155134, Oct 2018.

[69] Soonwon Choi, Christopher J. Turner, Hannes Pichler, Wen Wei Ho, Alex-ios A. Michailidis, Zlatko Papić, Maksym Serbyn, Mikhail D. Lukin, andDmitry A. Abanin. Emergent su(2) dynamics and perfect quantum many-body scars. Physical Review Letters, 122:220603, Jun 2019.

[70] Wen Wei Ho, Soonwon Choi, Hannes Pichler, and Mikhail D. Lukin. Periodicorbits, entanglement, and quantum many-body scars in constrained models:Matrix product state approach. Physical Review Letters, 122:040603, Jan2019.

[71] Paul Fendley, Krishnendu Sengupta, and Subir Sachdev. Competingdensity-wave orders in a one-dimensional hard-boson model’. Physical Re-view B, 69:075106, 2004.

[72] Federica M. Surace, Paolo P. Mazza, Giuliano Giudici, Alessio Lerose,Andrea Gambassi, and Marcello Dalmonte. Lattice gauge theories andstring dynamics in rydberg atom quantum simulators. Physical Review X,10:021041, May 2020.

[73] A. A. Michailidis, C. J. Turner, Z. Papić, D. A. Abanin, and M. Serbyn.Slow quantum thermalization and many-body revivals from mixed phasespace. Physical Review X, 10:011055, Mar 2020.

[74] Vedika Khemani, Chris R. Laumann, and Anushya Chandran. Signatures ofintegrability in the dynamics of rydberg-blockaded chains. Physical ReviewB, 99:161101, Apr 2019.

[75] Cheng-Ju Lin and Olexei I. Motrunich. Exact quantum many-body scarstates in the rydberg-blockaded atom chain. Physical Review Letters,122:173401, Apr 2019.

300

[76] Thomas Iadecola, Michael Schecter, and Shenglong Xu. Quantum many-body scars from magnon condensation. Physical Review B, 100:184312, Nov2019.

[77] Kieran Bull, Ivar Martin, and Z. Papić. Systematic construction ofscarred many-body dynamics in 1d lattice models. Physical Review Let-ters, 123:030601, Jul 2019.

[78] Sanjay Moudgalya, B. Andrei Bernevig, and Nicolas Regnault. QuantumMany-body Scars in a Landau Level on a Thin Torus. arXiv: 1906.05292,Jun 2019.

[79] Ana Hudomal, Ivana Vasić, Nicolas Regnault, and Zlatko Papić. Quantumscars of bosons with correlated hopping. arXiv: 1910.09526, Oct 2019.

[80] Shriya Pai and Michael Pretko. Dynamical scar states in driven fractonsystems. Physical Review Letters, 123:136401, Sep 2019.

[81] Bhaskar Mukherjee, Sourav Nandy, Arnab Sen, Diptiman Sen, and K. Sen-gupta. Collapse and revival of quantum many-body scars via floquet engi-neering. Physical Review B, 101:245107, Jun 2020.

[82] Neil J. Robinson, Andrew J. A. James, and Robert M. Konik. Signaturesof rare states and thermalization in a theory with confinement. PhysicalReview B, 99:195108, May 2019.

[83] Andrew J. A. James, Robert M. Konik, and Neil J. Robinson. Nonthermalstates arising from confinement in one and two dimensions. Physical ReviewLetters, 122:130603, Apr 2019.

[84] Asmi Haldar, Diptiman Sen, Roderich Moessner, and Arnab Das. Scars instrongly driven floquet matter: resonance vs emergent conservation laws.arXiv:1909.04064, 2019.

[85] Marko Znidaric. Coexistence of diffusive and ballistic transport in a simplespin ladder. Physical Review Letters, 110:070602, Feb 2013.

[86] Pablo Sala, Tibor Rakovszky, Ruben Verresen, Michael Knap, and FrankPollmann. Ergodicity breaking arising from hilbert space fragmentation indipole-conserving hamiltonians. Physical Review X, 10:011047, Feb 2020.

[87] Vedika Khemani, Michael Hermele, and Rahul Nandkishore. Localizationfrom hilbert space shattering: From theory to physical realizations. PhysicalReview B, 101:174204, May 2020.

301

[88] Sanjay Moudgalya, Abhinav Prem, Rahul Nandkishore, Nicolas Regnault,and B. Andrei Bernevig. Thermalization and its absence within krylovsubspaces of a constrained hamiltonian. arXiv: 1910.14048, 2019.

[89] Alessio Lerose, Federica Maria Surace, Paolo Pietro Mazza, Gabriele Per-fetto, Mario Collura, and Andrea Gambassi. Quasilocalized dynamics fromconfinement of quantum excitations. arXiv:1911.07877, 2019.

[90] Zhi-Cheng Yang, Fangli Liu, Alexey V. Gorshkov, and Thomas Iadecola.Hilbert-space fragmentation from strict confinement. Phys. Rev. Lett.,124:207602, May 2020.

[91] Álvaro M. Alhambra and Henrik Wilming. Revivals imply quantum many-body scars. arXiv: 1911.05637, 2019.

[92] Cheng-Ju Lin, Anushya Chandran, and Olexei I. Motrunich. Slow Thermal-ization of Exact Quantum Many-Body Scar States Under Perturbations.arXiv: 1910.07669, Oct 2019.

[93] Hongzheng Zhao, Joseph Vovrosh, Florian Mintert, and Johannes Knolle.Quantum many-body scars in optical lattices. arXiv: 2002.01746, 2020.

[94] Nicola Pancotti, Giacomo Giudice, J. Ignacio Cirac, Juan P. Garrahan,and Mari Carmen Bañuls. Quantum east model: localization, non-thermaleigenstates and slow dynamics. arXiv: 1910.06616, 2019.

[95] Kyungmin Lee, Ronald Melendrez, Arijeet Pal, and Hitesh J. Changlani.Exact three-colored quantum scars from geometric frustration. arXiv:2002.08970, 2020.

[96] Hyungwon Kim, Tatsuhiko N. Ikeda, and David A. Huse. Testing whether alleigenstates obey the eigenstate thermalization hypothesis. Physical ReviewE, 90:052105, Nov 2014.

[97] James R. Garrison and Tarun Grover. Does a single eigenstate encode thefull hamiltonian? Physical Review X, 8:021026, Apr 2018.

[98] Shriya Pai, Michael Pretko, and Rahul M. Nandkishore. Localization infractonic random circuits. Physical Review X, 9:021003, Apr 2019.

[99] F. Ritort and P. Sollich. Glassy dynamics of kinetically constrained models.Advances in Physics, 52(4):219–342, 2003.

[100] W. Beugeling, R. Moessner, and Masudul Haque. Finite-size scaling ofeigenstate thermalization. Physical Review E, 89:042112, Apr 2014.

302

[101] Sanjay Moudgalya, Trithep Devakul, D. P. Arovas, and S. L. Sondhi. Exten-sion of the eigenstate thermalization hypothesis to nonequilibrium steadystates. Physical Review B, 100:045112, Jul 2019.

[102] Freeman J Dyson. A brownian-motion model for the eigenvalues of a randommatrix. Journal of Mathematical Physics, 3(6):1191–1198, 1962.

[103] D Poilblanc, T Ziman, J Bellissard, F Mila, and G Montambaux. Pois-sonvs.GOE statistics in integrable and non-integrable quantum hamiltoni-ans. Europhysics Letters (EPL), 22(7):537–542, jun 1993.

[104] Vadim Oganesyan and David A Huse. Localization of interacting fermionsat high temperature. Physical Review B, 75(15):155111, 2007.

[105] Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux. Distribution of theRatio of Consecutive Level Spacings in Random Matrix Ensembles. PhysicalReview Letters, 110(8):084101, Feb 2013.

[106] J. Eisert, M. Cramer, and M. B. Plenio. Colloquium: Area laws for theentanglement entropy. Reviews of Modern Physics, 82:277–306, Feb 2010.

[107] Don N. Page. Average entropy of a subsystem. Physical Review Letters,71:1291–1294, Aug 1993.

[108] Ulrich Schollwöck. The density-matrix renormalization group in the age ofmatrix product states. Annals of Physics, 326(1):96 – 192, 2011. January2011 Special Issue.

[109] Frank Verstraete, Valentin Murg, and J Ignacio Cirac. Matrix productstates, projected entangled pair states, and variational renormalizationgroup methods for quantum spin systems. Advances in Physics, 57(2):143–224, 2008.

[110] Roman Orus. A practical introduction to tensor networks: Matrix productstates and projected entangled pair states. Annals of Physics, 349:117 – 158,2014.

[111] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac. Matrix productstate representations. Quantum Info. Comput., 7(5):401–430, July 2007.

[112] F. Verstraete and J. I. Cirac. Matrix product states represent ground statesfaithfully. Physical Review B, 73:094423, Mar 2006.

303

[113] J Ignacio Cirac, Didier Poilblanc, Norbert Schuch, and Frank Verstraete.Entanglement spectrum and boundary theories with projected entangled-pair states. Physical Review B, 83(24):245134, 2011.

[114] Ian Affleck, Tom Kennedy, Elliott H. Lieb, and Hal Tasaki. Valence bondground states in isotropic quantum antiferromagnets. Communications inMathematical Physics, 115(3):477–528, 1988.

[115] FDM Haldane. Nonlinear field theory of large-spin heisenberg antiferromag-nets: semiclassically quantized solitons of the one-dimensional easy-axis néelstate. Physical Review Letters, 50(15):1153, 1983.

[116] F Duncan M Haldane. Continuum dynamics of the 1-d heisenberg anti-ferromagnet: identification with the o (3) nonlinear sigma model. PhysicsLetters A, 93(9):464–468, 1983.

[117] Heng Fan, Vladimir Korepin, and Vwani Roychowdhury. Entanglement ina valence-bond solid state. Physical Review Letters, 93(22):227203, 2004.

[118] Hosho Katsura, Takaaki Hirano, and Yasuhiro Hatsugai. Exact analy-sis of entanglement in gapped quantum spin chains. Physical Review B,76(1):012401, 2007.

[119] Ying Xu, Hosho Katsura, Takaaki Hirano, and Vladimir E Korepin. En-tanglement and density matrix of a block of spins in aklt model. Journal ofStatistical Physics, 133(2):347–377, 2008.

[120] Frank Pollmann, Ari M Turner, Erez Berg, and Masaki Oshikawa. Entan-glement spectrum of a topological phase in one dimension. Physical ReviewB, 81(6):064439, 2010.

[121] Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen. Classification of gappedsymmetric phases in one-dimensional spin systems. Physical Review B,83(3):035107, 2011.

[122] Daniel P Arovas, Assa Auerbach, and FDM Haldane. Extended heisenbergmodels of antiferromagnetism: analogies to the fractional quantum hall ef-fect. Physical Review Letters, 60(6):531, 1988.

[123] Chanchal K Majumdar and Dipan K Ghosh. On next-nearest-neighborinteraction in linear chain. i. Journal of Mathematical Physics, 10(8):1388–1398, 1969.

[124] Daniel P. Arovas. Two exact excited states for the S = 1 AKLT chain.Physics Letters A, 137(7):431 – 433, 1989.

304

[125] Hui Li and F. D. M. Haldane. Entanglement spectrum as a generalizationof entanglement entropy: Identification of topological order in non-abelianfractional quantum hall effect states. Physical Review Letters, 101:010504,Jul 2008.

[126] Matthew B Hastings. An area law for one-dimensional quantum systems.Journal of Statistical Mechanics: Theory and Experiment, (08):P08024,2007.

[127] PN Bibikov. Three magnons in an isotropic s= 1 ferromagnetic chain asan exactly solvable non-integrable system. Journal of Statistical Mechanics:Theory and Experiment, (3):033109, 2016.

[128] Hirohito Kiwata and Yasuhiro Akutsu. Bethe-ansatz approximation for thes= 1 antiferromagnetic spin chain. Journal of the Physical Society of Japan,63(10):3598–3608, 1994.

[129] Tom Kennedy, Elliott H Lieb, and Hal Tasaki. A two-dimensional isotropicquantum antiferromagnet with unique disordered ground state. Journal ofStatistical Physics, 53(1):383–415, 1988.

[130] Martin Greiter, Stephan Rachel, and Dirk Schuricht. Exact results for su(3) spin chains: trimer states, valence bond solids, and their parent hamil-tonians. Physical Review B, 75(6):060401, 2007.

[131] Martin Greiter and Stephan Rachel. Valence bond solids for su (n) spinchains: exact models, spinon confinement, and the haldane gap. PhysicalReview B, 75(18):184441, 2007.

[132] Ulrich Schollwöck, Olivier Golinelli, and Thierry Jolicœur. S= 2 antiferro-magnetic quantum spin chain. Physical Review B, 54(6):4038, 1996.

[133] Jiadong Zang, Hong-Chen Jiang, Zheng-Yu Weng, and Shou-Cheng Zhang.Topological quantum phase transition in an s= 2 spin chain. Physical ReviewB, 81(22):224430, 2010.

[134] Hong-Chen Jiang, Stephan Rachel, Zheng-Yu Weng, Shou-Cheng Zhang,and Zhenghan Wang. Critical theory of the topological quantum phasetransition in an s= 2 spin chain. Physical Review B, 82(22):220403, 2010.

[135] Frank Pollmann, Erez Berg, Ari M Turner, and Masaki Oshikawa. Sym-metry protection of topological phases in one-dimensional quantum spinsystems. Physical Review B, 85(7):075125, 2012.

305

[136] Alexander V Turbiner. One-dimensional quasi-exactly solvable schrödingerequations. Physics Reports, 642:1–71, 2016.

[137] PN Bibikov. Bethe ansatz for two-magnon scattering states in 2d and 3dheisenberg–ising ferromagnets. Journal of Statistical Mechanics: Theoryand Experiment, 2018(4):043108, 2018.

[138] CJ Turner, AA Michailidis, DA Abanin, M Serbyn, and Z Papić. Weakergodicity breaking from quantum many-body scars. Nature Physics, 2018.

[139] Jutho Haegeman, Spyridon Michalakis, Bruno Nachtergaele, Tobias J Os-borne, Norbert Schuch, and Frank Verstraete. Elementary excitations ingapped quantum spin systems. Physical Review Letters, 111(8):080401,2013.

[140] Iztok Pizorn. Universality in entanglement of quasiparticle excitations.arXiv:1202.3336, 2012.

[141] Jutho Haegeman, Bogdan Pirvu, David J Weir, J Ignacio Cirac, Tobias JOsborne, Henri Verschelde, and Frank Verstraete. Variational matrix prod-uct ansatz for dispersion relations. Physical Review B, 85(10):100408, 2012.

[142] Jutho Haegeman, Tobias J. Osborne, and Frank Verstraete. Post-matrixproduct state methods: To tangent space and beyond. Physical Review B,88:075133, Aug 2013.

[143] Jutho Haegeman and Frank Verstraete. Diagonalizing transfer matrices andmatrix product operators: a medley of exact and computational methods.Annual Review of Condensed Matter Physics, 8:355–406, 2017.

[144] Valentin Zauner, Damian Draxler, Laurence Vanderstraeten, Matthias Deg-roote, Jutho Haegeman, Marek M Rams, Vid Stojevic, Norbert Schuch, andFrank Verstraete. Transfer matrices and excitations with matrix productstates. New Journal of Physics, 17(5):053002, 2015.

[145] V Zauner-Stauber, L Vanderstraeten, J Haegeman, IP McCulloch, andF Verstraete. Topological nature of spinons and holons: Elementary ex-citations from matrix product states with conserved symmetries. PhysicalReview B, 97(23):235155, 2018.

[146] Ronny Thomale, Stephan Rachel, B Andrei Bernevig, and Daniel P Arovas.Entanglement analysis of isotropic spin-1 chains. Journal of Statistical Me-chanics: Theory and Experiment, 2015(7):P07017, 2015.

306

[147] Mathis Friesdorf, Albert H Werner, W Brown, Volkher B Scholz, and JensEisert. Many-body localization implies that eigenvectors are matrix-productstates. Physical Review Letters, 114(17):170505, 2015.

[148] Vedika Khemani, Frank Pollmann, and SL Sondhi. Obtaining highly ex-cited eigenstates of many-body localized hamiltonians by the density matrixrenormalization group approach. Physical Review Letters, 116(24):247204,2016.

[149] Xiongjie Yu, David Pekker, and Bryan K Clark. Finding matrix productstate representations of highly excited eigenstates of many-body localizedhamiltonians. Physical Review Letters, 118(1):017201, 2017.

[150] Rubem Mondaini, Krishnanand Mallayya, Lea F Santos, and Marcos Rigol.Comment on” systematic construction of counterexamples to the eigenstatethermalization hypothesis”. arXiv:1711.06279, 2017.

[151] Naoto Shiraishi and Takashi Mori. Shiraishi and mori reply. Phys. Rev.Lett., 121:038902, Jul 2018.

[152] Oskar Vafek, Nicolas Regnault, and B. Andrei Bernevig. Entanglement ofExact Excited Eigenstates of the Hubbard Model in Arbitrary Dimension.SciPost Physics, 3:043, 2017.

[153] A Klümper, A Schadschneider, and J Zittartz. Matrix product ground statesfor one-dimensional spin-1 quantum antiferromagnets. Europhysics Letters(EPL), 24(4):293–297, nov 1993.

[154] K Totsuka and M Suzuki. Matrix formalism for the VBS-type models andhidden order. Journal of Physics: Condensed Matter, 7(8):1639–1662, feb1995.

[155] Mark Fannes, B Nachtergaele, and RF Werner. Exact antiferromagneticground states of quantum spin chains. EPL (Europhysics Letters), 10(7):633,1989.

[156] Mark Fannes, Bruno Nachtergaele, and Reinhard F Werner. Finitely cor-related states on quantum spin chains. Communications in mathematicalphysics, 144(3):443–490, 1992.

[157] Vahid Karimipour and Laleh Memarzadeh. Matrix product representationsfor all valence bond states. Physical Review B, 77:094416, Mar 2008.

307

[158] AK Kolezhuk, H-J Mikeska, and Shoji Yamamoto. Matrix-product-statesapproach to heisenberg ferrimagnetic spin chains. Physical Review B,55(6):R3336, 1997.

[159] Laurens Vanderstraeten, Michaël Mariën, Frank Verstraete, and JuthoHaegeman. Excitations and the tangent space of projected entangled-pairstates. Physical Review B, 92:201111, Nov 2015.

[160] Bogdan Pirvu, Jutho Haegeman, and Frank Verstraete. Matrix productstate based algorithm for determining dispersion relations of quantum spinchains with periodic boundary conditions. Physical Review B, 85(3):035130,2012.

[161] S Yi, ÖE Müstecaplıoğlu, Chang-Pu Sun, and Li You. Single-mode approx-imation in a spinor-1 atomic condensate. Physical Review A, 66(1):011601,2002.

[162] Daniel P Arovas. Simplex solid states of su (n) quantum antiferromagnets.Physical Review B, 77(10):104404, 2008.

[163] SM Girvin, AH MacDonald, and PM Platzman. Magneto-roton theory ofcollective excitations in the fractional quantum hall effect. Physical ReviewB, 33(4):2481, 1986.

[164] Frank Pollmann and Ari M Turner. Detection of symmetry-protected topo-logical phases in one dimension. Physical Review B, 86(12):125441, 2012.

[165] D. Pérez-García, M. M. Wolf, M. Sanz, F. Verstraete, and J. I. Cirac. Stringorder and symmetries in quantum spin lattices. Physical Review Letters,100:167202, Apr 2008.

[166] Laurens Vanderstraeten, Frank Verstraete, and Jutho Haegeman. Scatteringparticles in quantum spin chains. Physical Review B, 92(12):125136, 2015.

[167] Jutho Haegeman, Michaël Mariën, Tobias J Osborne, and Frank Verstraete.Geometry of matrix product states: Metric, parallel transport, and curva-ture. Journal of Mathematical Physics, 55(2):021902, 2014.

[168] MV Berry. Quantum scars of classical closed orbits in phase space. InProceedings of the Royal Society of London A: Mathematical, Physical andEngineering Sciences, volume 423, pages 219–231. The Royal Society, 1989.

[169] Sanjay Moudgalya, Trithep Devakul, C. W. von Keyserlingk, and S. L.Sondhi. Operator spreading in quantum maps. Phys. Rev. B, 99:094312,Mar 2019.

308

[170] Seulgi Ok, Kenny Choo, Christopher Mudry, Claudio Castelnovo, ClaudioChamon, and Titus Neupert. Topological many-body scar states in dimen-sions one, two, and three. Physical Review Research, 1:033144, Dec 2019.

[171] Edward O’Brien, Eric Vernier, and Paul Fendley. The “not-A”, SPT andPotts phases in an S3-invariant chain. arXiv: 1908.02767, 2019.

[172] Laurens Vanderstraeten, Jutho Haegeman, and Frank Verstraete. Tangent-space methods for uniform matrix product states. SciPost Physics LectureNotes, page 7, 2019.

[173] Marcel den Nijs and Koos Rommelse. Preroughening transitions in crystalsurfaces and valence-bond phases in quantum spin chains. Physical ReviewB, 40:4709–4734, Sep 1989.

[174] Tom Kennedy and Hal Tasaki. Hidden z2×z2 symmetry breaking in haldane-gap antiferromagnets. Physical Review B, 45:304–307, Jan 1992.

[175] M Oshikawa. Hidden z2∗z2symmetry in quantum spin chains with arbitraryinteger spin. Journal of Physics: Condensed Matter, 4(36):7469–7488, sep1992.

[176] G Schutz. ‘Duality twisted’ boundary conditions in n-state Potts models.Journal of Physics A: Mathematical and General, 26(18):4555, 1993.

[177] J B Parkinson. On the integrability of the s=1 quantum spin chain withpure biquadratic exchange. Journal of Physics C: Solid State Physics,20(36):L1029–L1032, dec 1987.

[178] Michael N. Barber and Murray T. Batchelor. Spectrum of the biquadraticspin-1 antiferromagnetic chain. Physical Review B, B40:4621–4626, 1989.

[179] Elisa Ercolessi, Dott Davide Vodola, and Ferri Silvia. Analysis of the spec-trum of the spin-1 biquadratic antiferromagnetic chain, 2014.

[180] Norbert Schuch, Ignacio Cirac, and David Pérez-García. Peps as groundstates: Degeneracy and topology. Annals of Physics, 325(10):2153 – 2192,2010.

[181] Naoto Shiraishi. Connection between quantum-many-body scars and theaffleck–kennedy–lieb–tasaki model from the viewpoint of embedded hamil-tonians. Journal of Statistical Mechanics: Theory and Experiment,2019(8):083103, aug 2019.

309

[182] Daniel K. Mark, Cheng-Ju Lin, and Olexei I. Motrunich. Exact eigenstatesin the lesanovsky model, proximity to integrability and the pxp model, andapproximate scar states. Physical Review B, 101:094308, Mar 2020.

[183] Igor Lesanovsky. Many-body spin interactions and the ground state of adense rydberg lattice gas. Physical Review Letters, 106(2):025301, 2011.

[184] Vedika Khemani, Chris R Laumann, and Anushya Chandran. Signatures ofintegrability in the dynamics of rydberg-blockaded chains. Physical ReviewB, 99(16):161101, 2019.

[185] Chen Ning Yang. η pairing and off-diagonal long-range order in a hubbardmodel. Physical Review letters, 63(19):2144, 1989.

[186] Shoucheng Zhang. Pseudospin symmetry and new collective modes of thehubbard model. Physical Review Letters, 65:120–122, Jul 1990.

[187] Chen Ning Yang and Shoucheng Zhang. So(4) symmetry in a hubbardmodel. Modern Physics Letters B, 4(11):759–766, 1990.

[188] Chen Ning Yang. Remarks and generalizations about su2×su2 symmetry ofhubbard models. Physics Letters A, 161(3):292 – 294, 1991.

[189] Shun-Qing Shen and Zhao-Ming Qiu. Exact demonstration of off-diagonallong-range order in the ground state of a hubbard model. Physical ReviewLetters, 71:4238–4240, Dec 1993.

[190] Fabian H. L. Essler. Extended hubbard models, eta-pairing and supersym-metry. Journal of Low Temperature Physics, 99(3):415–420, 1995.

[191] Jan de Boer, Vladimir E. Korepin, and Andreas Schadschneider. η pairing asa mechanism of superconductivity in models of strongly correlated electrons.Physical Review Letters, 74:789–792, Jan 1995.

[192] G Albertini, V E Korepin, and A Schadschneider. XXZ model as an effectivehamiltonian for generalized hubbard models with broken eta -symmetry.Journal of Physics A: Mathematical and General, 28(10):L303–L309, may1995.

[193] Andreas Schadschneider. Superconductivity in an exactly solvable hubbardmodel with bond-charge interaction. Physical Review B, 51:10386–10391,Apr 1995.

[194] Heng Fan. The fermion-ladder models: extensions of the hubbard modelwith eta-pairing. 32(48):L509–L513, nov 1999.

310

[195] Heng Fan and Seth Lloyd. Entanglement and off-diagonal long-range orderof an eta-pairing state. Journal of Physics A: Mathematical and General,38(23):5285–5292, may 2005.

[196] Hui Zhai. Two generalizations of η pairing in extended hubbard models.Physical Review B, 71:012512, Jan 2005.

[197] Kai Li. Exact su(2) symmetry and -pairing ground states in interactingfermion models with spin-orbit coupling. arXiv: 1901.06914, 2019.

[198] Fabian HL Essler, Holger Frahm, Frank Göhmann, Andreas Klümper, andVladimir E Korepin. The one-dimensional Hubbard model. Cambridge Uni-versity Press, 2005.

[199] Xiongjie Yu, Di Luo, and Bryan K. Clark. Beyond many-body localizedstates in a spin-disordered hubbard model. Physical Review B, 98:115106,Sep 2018.

[200] A. O. Barut and A. Böhm. Dynamical groups and mass formula. PhysicalReview, 139:B1107–B1112, Aug 1965.

[201] Y. Dothan, M. Gell-Mann, and Y. Ne’eman. Series of hadron energy levelsas representations of non-compact groups. Physics Letters, 17(2):148 – 151,1965.

[202] Bohm Arno, Asim Orhan Barut, et al. Dynamical groups and spectrumgenerating algebras, volume 1. World Scientific, 1988.

[203] Allan I Solomon and K A Penson. Coherent pairing states for the hubbardmodel. Journal of Physics A: Mathematical and General, 31(18):L355–L360,may 1998.

[204] Bruno Gruber and Takaharu Otsuka. Symmetries in science VII: spectrum-generating algebras and dynamic symmetries in physics. Springer Science &Business Media, 2012.

[205] A. Leviatan. Partial dynamical symmetries. Progress in Particle and NuclearPhysics, 66(1):93 – 143, 2011.

[206] C. D. Batista. Canted spiral: An exact ground state of xxz zigzag spinladders. Physical Review B, 80:180406, Nov 2009.

[207] Jurriaan Wouters, Hosho Katsura, and Dirk Schuricht. Exact ground statesfor interacting kitaev chains. Physical Review B, 98:155119, Oct 2018.

311

[208] Thomas Iadecola and Marko Znidaric. Exact localized and ballistic eigen-states in disordered chaotic spin ladders and the fermi-hubbard model. Phys-ical Review Letters, 123:036403, Jul 2019.

[209] Rubem Mondaini and Marcos Rigol. Many-body localization and thermal-ization in disordered hubbard chains. Physical Review A, 92:041601, Oct2015.

[210] P. Prelovšek, O. S. Barišić, and M. Žnidarič. Absence of full many-bodylocalization in the disordered hubbard chain. Physical Review B, 94:241104,Dec 2016.

[211] S. S. Kondov, W. R. McGehee, W. Xu, and B. DeMarco. Disorder-inducedlocalization in a strongly correlated atomic hubbard gas. Physical ReviewLetters, 114:083002, Feb 2015.

[212] Maciej Kozarzewski, Peter Prelovšek, and Marcin Mierzejewski. Spin subdif-fusion in the disordered hubbard chain. Physical Review Letters, 120:246602,Jun 2018.

[213] Marcin Mierzejewski, Maciej Kozarzewski, and Peter Prelovšek. Countinglocal integrals of motion in disordered spinless-fermion and hubbard chains.Physical Review B, 97:064204, Feb 2018.

[214] Alexander Seidel, Henry Fu, Dung-Hai Lee, Jon Magne Leinaas, and JoelMoore. Incompressible quantum liquids and new conservation laws. PhysicalReview Letters, 95:266405, Dec 2005.

[215] Emil J. Bergholtz and Anders Karlhede. ’One-dimensional’ theory of thequantum Hall system. Journal of Statistical Mechanics: Theory and Exper-iment, 2006(4):L04001, Apr 2006.

[216] E. J. Bergholtz and A. Karlhede. Quantum hall system in tao-thouless limit.Physical Review B, 77:155308, Apr 2008.

[217] Ching Hua Lee, Zlatko Papić, and Ronny Thomale. Geometric constructionof quantum hall clustering hamiltonians. Physical Review X, 5:041003, Oct2015.

[218] Z. Papić. Solvable models for unitary and nonunitary topological phases.Physical Review B, 90:075304, Aug 2014.

[219] E. H. Rezayi and F. D. M. Haldane. Laughlin state on stretched andsqueezed cylinders and edge excitations in the quantum hall effect. PhysicalReview B, 50:17199–17207, Dec 1994.

312

[220] Masaaki Nakamura, Zheng-Yuan Wang, and Emil J. Bergholtz. Exactlysolvable fermion chain describing a ν = 1/3 fractional quantum hall state.Physical Review Letters, 109:016401, Jul 2012.

[221] Gregory H. Wannier. Dynamics of band electrons in electric and magneticfields. Reviews of Modern Physics, 34:645–655, Oct 1962.

[222] David Emin and C. F. Hart. Existence of wannier-stark localization. Phys-ical Review B, 36:7353–7359, Nov 1987.

[223] Evert van Nieuwenburg, Yuval Baum, and Gil Refael. From bloch oscilla-tions to many-body localization in clean interacting systems. Proceedingsof the National Academy of Sciences, 116(19):9269–9274, 2019.

[224] M. Schulz, C. A. Hooley, R. Moessner, and F. Pollmann. Stark many-bodylocalization. Physical Review Letters, 122:040606, Jan 2019.

[225] Zheng-Yuan Wang, Shintaro Takayoshi, and Masaaki Nakamura. Spin-chaindescription of fractional quantum hall states in the jain series. PhysicalReview B, 86:155104, Oct 2012.

[226] Rahul M. Nandkishore and Michael Hermele. Fractons. Annual Review ofCondensed Matter Physics, 10(1):295–313, 2019.

[227] Michael Pretko. Subdimensional particle structure of higher rank u(1) spinliquids. Physical Review B, 95:115139, Mar 2017.

[228] M. Fremling, Cécile Repellin, Jean-Marie Stéphan, N. Moran, J. K. Slinger-land, and Masudul Haque. Dynamics and level statistics of interactingfermions in the lowest Landau level. New Journal of Physics, 20(10):103036,Oct 2018.

[229] Emil J Bergholtz and Anders Karlhede. Half-filled lowest landau level on athin torus. Physical Review Letters, 94(2):026802, 2005.

[230] Elliott H. Lieb and Werner Liniger. Exact analysis of an interacting bosegas. i. the general solution and the ground state. Physical Review, 130:1605–1616, May 1963.

[231] Giuseppe De Tomasi, Daniel Hetterich, Pablo Sala, and Frank Pollmann.Dynamics of strongly interacting systems: From fock-space fragmentationto many-body localization. Physical Review B, 100:214313, Dec 2019.

313

[232] E Urban, Todd A Johnson, T Henage, L Isenhower, DD Yavuz, TG Walker,and M Saffman. Observation of rydberg blockade between two atoms. Na-ture Physics, 5(2):110, 2009.

[233] Igor Lesanovsky and Hosho Katsura. Interacting fibonacci anyons in a ry-dberg gas. Physical Review A, 86(4):041601, 2012.

[234] Jutho Haegeman, J Ignacio Cirac, Tobias J Osborne, Iztok Pižorn, HenriVerschelde, and Frank Verstraete. Time-dependent variational principle forquantum lattices. Physical Review Letters, 107(7):070601, 2011.

[235] R Moessner and SL Sondhi. Ising models of quantum frustration. PhysicalReview B, 63(22):224401, 2001.

[236] B Andrei Bernevig and FDM Haldane. Model fractional quantum hall statesand jack polynomials. Physical Review Letters, 100(24):246802, 2008.

[237] B Andrei Bernevig and N Regnault. Anatomy of abelian and non-abelianfractional quantum hall states. Physical Review Letters, 103(20):206801,2009.

[238] Hyungwon Kim and David A Huse. Ballistic spreading of entanglement ina diffusive nonintegrable system. Physical Review Letters, 111(12):127205,2013.

[239] Bogdan Pirvu, Valentin Murg, J Ignacio Cirac, and Frank Verstraete. Matrixproduct operator representations. New Journal of Physics, 12(2):025012,2010.

[240] Ian P McCulloch. From density-matrix renormalization group to matrixproduct states. Journal of Statistical Mechanics: Theory and Experiment,2007(10):P10014, 2007.

[241] Frank Verstraete, Juan J Garcia-Ripoll, and Juan Ignacio Cirac. Matrixproduct density operators: simulation of finite-temperature and dissipativesystems. Physical Review Letters, 93(20):207204, 2004.

[242] Gregory M Crosswhite and Dave Bacon. Finite automata for caching inmatrix product algorithms. Physical Review A, 78(1):012356, 2008.

[243] Johannes Motruk, Michael P Zaletel, Roger SK Mong, and Frank Pollmann.Density matrix renormalization group on a cylinder in mixed real and mo-mentum space. Physical Review B, 93(15):155139, 2016.

314

[244] Roger Fletcher and Danny C Sorensen. An algorithmic derivation of thejordan canonical form. The American Mathematical Monthly, 90(1):12–16,1983.

[245] Padraic Bartlett. Lecture 8: The Jordan Canonical Form, UCSB Math108B, 2013.

[246] Sergey Bravyi, David P. DiVincenzo, and Daniel Loss. Schriefferâwolfftransformation for quantum many-body systems. Annals of Physics,326(10):2793 – 2826, 2011.

[247] FDM Haldane. Many-particle translational symmetries of two-dimensionalelectrons at rational landau-level filling. Physical Review Letters,55(20):2095, 1985.

[248] F Duncan M Haldane and Edward H Rezayi. Periodic laughlin-jastrowwave functions for the fractional quantized hall effect. Physical Review B,31(4):2529, 1985.

[249] Eduardo Fradkin. Field theories of condensed matter physics. CambridgeUniversity Press, 2013.

[250] F Duncan M Haldane. Fractional quantization of the hall effect: a hierarchyof incompressible quantum fluid states. Physical Review Letters, 51(7):605,1983.

[251] SA Trugman and S Kivelson. Exact results for the fractional quantum halleffect with general interactions. Physical Review B, 31(8):5280, 1985.

[252] Pietro Rotondo, Luca Guido Molinari, Piergiorgio Ratti, and Marco Gher-ardi. Devil’s staircase phase diagram of the fractional quantum hall effectin the thin-torus limit. Physical Review Letters, 116(25):256803, 2016.

[253] Izrail Solomonovich Gradshteyn and Iosif Moiseevich Ryzhik. Table of in-tegrals, series, and products. Academic press, 2014.

315