Interactions in Topological Matter · Interactions in Topological Matter Christopher Mudry1 1Paul...

Interactions in Topological Matter

Christopher Mudry1

1Paul Scherrer Institute, Switzerland

Harish-Chandra Research Institute, Allahabad, February 9-20 2015

C. Mudry (PSI) Interactions in Topological Matter 1 / 108

Outline

1 IntroductionTutorials

2 The tenfold way in quasi-one-dimensional space

3 Fractionalization from Abelian bosonizationTutorials

4 Stability analysis for the edge theory in the symmetry class AII

5 Construction of two-dimensional topological phases from coupledwires


1931: Dirac introduces topology in physics

Paul Dirac, “Quantised Singularities in the Electromagnetic Field,”Proc. Roy. Soc. (London) A 133, 60 (1931).


1932-1939: Tamm and Schockley surface states

I. Tamm (1932), “On the possible bound states of electrons on acrystal surface,” Phys. Z. Soviet Union 1, 733 (1932).

W. Shockley, “On the Surface States Associated with a PeriodicPotential,” Phys. Rev. 56, 317 (1939).

P-Y. Chang, C. Mudry,and S. Ryu 2014

Direct sum of apx + ipy and of apx − ipy BdGsuperconductor in acylindrical geometry.


1950’s: Anderson localization, Dyson’s exception

P. W. Anderson, “Absence ofDiffusion in Certain RandomLattices,” Phys. Rev. 109, 1492(1957).

F. J. Dyson, “The Dynamics of aDisordered Linear Chain,”Phys. Rev. 92, 1331 (1953).


1963: The threefold way for random matrices

F. J. Dyson, “The Threefold Way. Algebraic Structure of SymmetryGroups and Ensembles in Quantum Mechanics,” J. Math. Phys. 3,1199 (1962):

P(θ1, · · · , θN) ∝∏

1≤j<k≤N

∣∣∣eiθj − eiθk

∣∣∣β , β = 1,2,4.


1971-1974: Berezinski-Kosterlitz-Thouless transition

Topology acquires a mainstream status in physics as of 1973 with thedisovery of Berezinskii and of Kosterlitz and Thousless that topologicaldefects in magnetic classical textures can drive a phase transition.

V. L. Berezinskii, “Destruction of Long-range Order inOne-dimensional and Two-dimensional Systems having aContinuous Symmetry Group I. Classical Systems,” SovietJournal of Experimental and Theoretical Physics, 32 493, (1971).J. M. Kosterlitz and D. J. Thouless, “Ordering, metastability andphase transitions in two-dimensional systems,”, J. Phys. C, 6,1181 (1973).J. M. Kosterlitz, “The critical properties of the two-dimensional xymodel,” J. Phys. C, 7, 1046 (1974).


1976: Jackiw and Rebbi introduce Fermion numberfractionalizationR. Jackiw and C. Rebbi, “Solitons with fermion number 1/2,” Phys. Rev.D 13, 3398 (1976): Dirac equation with a single point defect in abackground field supports a single zero mode that carries the fermionnumber 1/2.W. P. Su, J. R. Schrieffer, and H. J. Heeger, “Soliton excitations inpolyacetylene,” Phys. Rev. B 22, 2099 (1980): Propose polaycetyleneas a realization of fermion fractionalization

��

��

��

��

��

��

+π/2−π/2

k

ε (k)


1981: Nielsen-Ninomiya theorem

H. B. Nielsen and M. Ninomiya, “A no-go theorem for regularizingchiral fermions,” Phys. Lett. B105, 219 (1981): The Nielsen-Ninomiyatheorem is a no-go theorem that prohibits defining a theory of chiralfermions on a lattice in odd-dimensional space.


1980’s: The Quantum Hall EffectK. von Klitzing, G. Dorda, and M. Pepper, “New Method forHigh-Accuracy Determination of the Fine-Structure Constant Based onQuantized Hall,” Phys. Rev. Lett. 45, 494 (1980).Graphene deposited on SiO2/Si, T =1.6 K and B=9 T (inset T =30 mK):ν = ±2,±6,±10, · · · = ±2(2n + 1), n ∈ N

after Zhang et al., Nature 438, 201 (2005).


The Integer Quantum Hall Effect

εh ωm c

DOSσxy

Without disorder

m=−1

m= 0

m= 1

n ε h ωm c

DOS

With disorder

m= 1

m= 0

m=−1

but no interactions2

6

−2

−6

σ [e^2/h]xy

orεF

n

At integer fillings of the Landau levels, the noninteracting ground stateis unique and the screened Coulomb interaction Vint can be treatedperturbatively, as long as transitions between Landau levels or outsidethe confining potential Vconf along the magnetic field are suppressed bythe single-particle gaps:

Vint � ~ωc � Vconf, ωc = e B/(m c).


1982: The Fractional Quantum Hall Effect

D. C. Tsui, H. L. Stormer, and A. C. Gossard, “Two-DimensionalMagnetotransport in the Extreme Quantum Limit,” Phys. Rev. Lett. 48,1559 (1982).

At fractional fillings of a Landau level, rs is effectively∞: A landau levelis a massively degenerate flat band of single-particle states.

Naively, one would expect a Wigner crystal (or more exotic groundstates with broken symmetry) to be selected by the interaction out ofall possible degenerate Slater determinants.

Instead, for “magic” filling fractions, featureless (i.e., liquid like) groundstates are selected by the screened Coulomb interaction.

For example, whenever 1/ν is an odd integer, the featureless groundstate is an incompressible ground state called a Laughlin state.


Distinctive signatureThe conductivity tensor is given by the classical Drude formula

limτ→∞

j =

(0 +

(B RH

)−1

−(B RH

)−1 0

)E , R−1

H := −n e c.

in the ballistic regime when Galilean invariance is not broken.

In the presence of moderate static disorder, all but one single-particlesare localized in a Landau level whereas many-body groundstates suchas the Wigner crystal are pinned.

In the presence of moderate static disorder themagic filling fractions turn into plateaus at which

σxx = 0, σxy = ν × e2

h

as a function of B for fixed n.


1981-1982: Laughlin and Halperin introduce thebulk-edge correspondence in the Quantum Hall Effect

Laughlin 1981: The Hall conductivitymust be rational and if it is not aninteger, the ground state manifold mustbe degenerate and support fractionallycharged excitations

B

x

y

z

(a) (b)

Halperin 1982: Chiral edges are immune to backscattering within eachtraffic lane

Integer Quantum Hall Effect Fractional Quantum Hall Effect


1982: TKNN relate linear response to topology in theQuantum Hall EffectThe Hall conductance is proportional to the first Chern number

C = − i2π

2π∫0

dφ

2π∫0

dϕ[⟨

∂Ψ

∂φ

∣∣∣∣ ∂Ψ

∂ϕ

⟩−⟨∂Ψ

∂ϕ

∣∣∣∣ ∂Ψ

∂φ

⟩]

with Ψ the many-body ground state obeying twisted boundaryconditions.D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall Conductance ina Two-Dimensional Periodic Potential,” Phys. Rev. Lett. 49, 405 (1982).

J. E. Avron, R. Seiler and B. Simon, “Homotopy and Quantization in Condensed Matter Physics,”Phys. Rev. Lett. 51, 51 (1983); “Holonomy, the Quantum Adiabatic Theorem, and Berry’s Phase,”B. Simon, ibid. 51, 2167 (1983).

Q. Niu and D. J. Thouless, “Quantised adiabatic charge transport in the presence of substratedisorder and many-body interaction,” J. Phys. A 17, 2453 (1984).

Q. Niu, D. J. Thouless, and Y. S. Wu, “Quantized Hall conductance as a topological invariant,”

Phys. Rev. B 31, 3372 (1985).


1983-1985: Khmelnitskii and Pruisken introduce thescaling theory of the Integer Quantum Hall Effect

A topological term modifies the scaling analysis of the gang of four:Khmelnitskii 1983 Pruisken 1985

ln g 0

1

−1

d ln g d ln L

d=2

φ=0

φ=π

0 1 2 3σ

σxx

xy


1983-1984: Haldane introduces the θ term for spinchains and Witten achieves non-Abelian bosonization

F. D. M. Haldane, “Nonlinear Field Theory of Large-Spin HeisenbergAntiferromagnets: Semiclassically Quantized Solitons of theOne-Dimensional Easy-Axis Néel State,” Phys. Rev. Lett. 50, 1153(1983), “Continuum dynamics of the 1-D Heisenberg antiferromagnet:Identification with the O(3) nonlinear sigma model,” Phys. Lett. A 93,464 (1983) : Topological θ term modifies the RG flow in thetwo-dimensional O(3) non-linear-sigma model.

g g

O(3) NLSM O(3) NLSM+

SU(2)

θ=π

1

E. Witten, “Nonabelian bosonization in two dimensions,” Commun.Math. Phys. 92, 455 (1984): Non-Abelian bosonization.


1988: Haldane model

F. D. M. Haldane, “Model for a Quantum Hall Effect without LandauLevels: Condensed-Matter Realization of the ‘Parity Anomaly’,” Phys.Rev. Lett. 61, 2015 (1988).


1994: Random Dirac fermions in two-dimensionalspaceA. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein,“Integer quantum Hall transition: An alternative approach and exactresults,” Phys. Rev. B 50, 7526 (1994): Effects of static disorder on asingle Dirac fermion in two-dimensional space.

AIII

AII

D

IQHE


1997: The tenfold way for random matricesA. Altland and M. R. Zirnbauer, “Nonstandard symmetry classes inmesoscopic normal-superconducting hybrid structures,” Phys. Rev. B55, 1142 (1997).

Cartan label T C S Hamiltonian G/H (ferm. NLSM)

A (unitary) 0 0 0 U(N) U(2n)/U(n)× U(n)

AI (orthogonal) +1 0 0 U(N)/O(N) Sp(2n)/Sp(n)× Sp(n)

AII (symplectic) −1 0 0 U(2N)/Sp(2N) O(2n)/O(n)× O(n)

AIII (ch. unit.) 0 0 1 U(N + M)/U(N)× U(M) U(n)

BDI (ch. orth.) +1 +1 1 O(N + M)/O(N)× O(M) U(2n)/Sp(2n)

CII (ch. sympl.) −1 −1 1 Sp(N + M)/Sp(N)× Sp(M) U(2n)/O(2n)

D (BdG) 0 +1 0 SO(2N) O(2n)/U(n)

C (BdG) 0 −1 0 Sp(2N) Sp(2n)/U(n)

DIII (BdG) −1 +1 1 SO(2N)/U(N) O(2n)

CI (BdG) +1 −1 1 Sp(2N)/U(N) Sp(2n)

The column entitled “Hamiltonian” lists, for each of the ten symmetry classes, the symmetric space of which the quantum

mechanical time-evolution operator exp(itH) is an element. The last column entitled “G/H (ferm. NLσM)” lists the (compact

sectors of the) target space of the NLσM describing Anderson localization physics at long wavelength in this given symmetry

class. C. Mudry (PSI) Interactions in Topological Matter 20 / 108

1998-2000: Brouwer et al. establish that there are fivesymmetry class that are quantum critical in disorderedquasi-one-dimensional wires

The “radial coordinate” of thetransfer matrixM from the Tablebelow makes a Brownian motionon an associated symmetric space.

Class TRS SRS mo ml D M H δg 〈− ln g〉 ρ(ε) for 0 < ετc � 1O Yes Y 1 1 2 CI AI −2/3 2L/(γ`) ρ0U No Y(N) 2 1 2(1) AIII A 0 2L/(γ`) ρ0S Y N 4 1 2 DIII AII +1/3 2L/(γ`) ρ0chO Y Y 1 0 2 AI BDI 0 2moL/(γ`) ρ0| ln |ετc ||chU N Y(N) 2 0 2(1) A AIII 0 2moL/(γ`) πρ0|ετc ln |ετc ||chS Y N 4 0 2 AII CII 0 2moL/(γ`) (πρ0/3)|(ετc )3 ln |ετc ||CI Y Y 2 2 4 C CI −4/3 2ml L/(γ`) (πρ0/2)|ετc |C N Y 4 3 4 CII C −2/3 2ml L/(γ`) ρ0|ετc |2

DIII Y N 2 0 2 D DIII +2/3 4√

L/(2πγ`) πρ0/|ετc ln3 |ετc ||D N N 1 0 1 BDI D +1/3 4

√L/(2πγ`) πρ0/|ετc ln3 |ετc ||


2000: Read and Green introduce the chiral p-wavetopological superconductor

N. Read and D. Green, “Paired states of fermions in two dimensionswith breaking of parity and time-reversal symmetries and the fractionalquantum Hall effect,” Phys. Rev. B 61, 10267 (2000).


2005: Kane and Mele introduce the strong Z2topological insulatorThe spin-orbit coupling is ignored in the QHE as the breaking oftime-reversal symmetry provides the dominant energy scale.C. L. Kane and E. J. Mele, “Z2 Topological Order and the QuantumSpin Hall Effect,” Phys. Rev. Lett. 95, 146802 (2005): Combine a pairof time-reversed Haldane models with a small Rashba coupling andfind protected helical edge states.

kx

E

ky !

E0

k0

EF

S

(for a Bix Pb1−x /Ag(111) surface alloy, say)


2008: The tenfold way for topological insulators andsuperconductorsSchnyder, Ryu, Furusaki, and Ludwig 2008 and 2010 ; Kitaev 2008

complex case:

Cartan\d 0 1 2 3 4 5 6 7 8 9 10 11 · · ·A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0 · · ·

AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z · · ·

real case:

Cartan\d 0 1 2 3 4 5 6 7 8 9 10 11 · · ·AI Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 · · ·

BDI Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 · · ·D Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 · · ·

DIII 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z · · ·AII 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 · · ·CII 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 · · ·C 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 · · ·CI 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z · · ·


2011: Fractional Chern and fractional Z2 topologicalinsulators

T. Neupert, L. Santos, C. Chamon,and C. Mudry, “Fractional QuantumHall States at Zero Magnetic Field,”Phys. Rev. Lett. 106, 236804(2011) and T. Neupert, L. Santos,S. Ryu, C. Chamon, and C. Mudry,“Fractional topological liquids withtime-reversal symmetry and theirlattice realization,” Phys. Rev. B 84,165107 (2011).

λ

a)

b)

0 10 20

3-fold

9-fold

0

1

0.5

0 1 2 3 U/V

0

0.1

E/V

γx

20

0.2

U/V = 0, λ = 0c)

γx

20 γx

20

d)U/V = 0, λ = 1 U/V = 3, λ = 1

∋


The goals of these lectures are the following.First, we would like to rederive the tenfold way for non-interactingfermions in the presence of local interactions and static localdisorder.Second, we would like to decide if interactions between fermionscan produce topological phases of matter with protected boundarystates that are not captured by the tenfold way.

We will apply this program in two-dimensional space.


Main result 2

TABLE I. (Color online) Realization of a two-dimensional array of quantum wires in each symmetry class of the tenfold way.

Θ2 Π2 C2 Short-range entangled (SRE) topological phase Long-range entangled (LRE) topological phase

A 0 0 0 Z ��

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AIII 0 0 + NONE

AII − 0 0 Z2��

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��

��

��

DIII − + + Z2

��

��

��

��

��

D 0 + 0 Z ��

��

��

��

BDI + + + NONE

AI + 0 0 NONE

CI + − + NONE

C 0 − 0 Z��

��

��

��

��

��

CII − − + NONE


Organization of the lectures

Introduction X

The tenfold way in quasi-one-dimensional space

Fractionalization from Abelian bosonization

Stability analysis for the edge theory in the symmetry class AII

Construction of two-dimensional topological phases from coupledwires


Tutorial on Galilean transformation

A Galilean transformation is a transformation to a new frame ofreference that leaves the time difference t1 − t2 and space separation|x1 − x2| unchanged as well as the equation of motion

m x = 0

of a free particle form invariant. It is given by

t ′ = t + a, x ′ = Ox + v t + w , a ∈ R, v ,w ∈ R3, O ∈ O(3).

The transformation law of momentum p ≡ mx and kinetic energyEkin ≡ (1/2)mx2 under a Galilean transformation are

p′ = Op + mv , E ′kin = Ekin + m (Ox) · v + (1/2)m v2.


Tutorial on reversal of time:Reversal of time

A Galilean transformation can be composed with reversal of time

t = −t ′, x = x ′.

Under reversal of time, the momentum

p := m x = −p′

is odd and so is the angular momentum

L := x ∧ p = −L′.


Tutorial on reversal of time:The case of classical electromagnetism

Maxwell equations are separated into the pair of homogeneousequations

∇ ∧ B = 0, ∇ ∧ E +1c∂

∂tB = 0,

and into the pair of inhomogeneous equations

∇ · E = 4π ρ, ∇ ∧ B − 1c∂

∂tE =

4πc

j .

They are invariant under the transformation laws

t = −t ′, E(x ,−t ′) = +E(x , t ′), B(x ,−t ′) = −B(x , t ′),ρ(x ,−t ′) = +ρ(x , t ′), j(x ,−t ′) = −j(x , t ′).


Tutorial on reversal of time:The case of quantum mechanics for a spinless particleWe start with the non-relativistic Schrödinger equation for a spinlessparticle

i~∂

∂tψ(x , t) =

{1

2m

[p − e

cA(x , t)

]2+ e A0(x , t) + V (x , t)

}ψ(x , t).

How to implement reversal of time? We demand that

p :=~i∂

∂xis odd under reversal of time

t = −t ′, x = +x ′.

This is achieved by defining

p = K p′ K = −p′

where K represents the antilinear operation of complex conjugation.C. Mudry (PSI) Interactions in Topological Matter 32 / 108

We conclude that

i~∂

∂tψ(x , t) =

{1

2m

[p − e

cA(x , t)

]2+ e A0(x , t) + V (x , t)

}ψ(x , t)

is invariant under the transformations

t = −t ′, x = +x ′,ψ∗(+x ′,−t ′) ≡ Kψ(+x ′,−t ′) = ψ(x ′, t ′),p∗ ≡ K p K = −p,A(+x ′,−t ′) = −A(x ′, t ′),A0(+x ′,−t ′) = +A0(x ′, t ′), V (+x ′,−t ′) = +V (x ′, t ′).


Tutorial on reversal of time:The case of quantum mechanics for a spin-1/2 particleWe start with the Pauli equation for a spin-1/2 particle

i~ ∂∂t

Ψ(x , t) =

{1

2m

[σ ·(

p − ec

A(x , t))]2

+ σ0 [e A0(x , t) + V (x , t)]

}Ψ(x , t).

How to implement reversal of time? We demand that

σ0 p := σ0~i∂

∂xand σ ≡ (σ1, σ2, σ3)

are both odd under reversal of time

t = −t ′, x = +x ′.

This is achieved by defining

σ0 p = (σ2 K)σ0 p′ (Kσ2) = −σ0 p′ and σ = (σ2 K)σ′ (Kσ2) = −σ′,

where K represents the antilinear operation of complex conjugation.C. Mudry (PSI) Interactions in Topological Matter 34 / 108

We conclude that

i~ ∂∂t

Ψ(x , t) =

{1

2m

[σ ·(

p − ec

A(x , t))]2

+ σ0 [e A0(x , t) + V (x , t)]

}Ψ(x , t).

is invariant under the transformations

t = −t ′, x = +x ′,σ2 Ψ∗(+x ′,−t ′) ≡ (σ2 K) Ψ(+x ′,−t ′) = Ψ(x ′, t ′),(σ2 K) σ (Kσ2) = −σ, σ0 p∗ ≡ (σ2 K) σ0 p (Kσ2) = −σ0 p,A(+x ′,−t ′) = −A(x ′, t ′),A0(+x ′,−t ′) = +A0(x ′, t ′), V (+x ′,−t ′) = +V (x ′, t ′).


Tutorial on one-dimensional Dirac Hamiltonians

Let the Bloch Hamiltonian for one spinless fermion be

H(k) := −2t cos k , −π ≤ k < +π,

(uniform nearest-neighbor hopping with t > 0).Linearization of this dispersion about the two Fermi points ±kF with−π < kF < +π gives the rank-two Dirac Hamiltonian

HD := −τ3 i∂

∂x(~ ≡ 1, vF = 2t | sin kF| ≡ 1).

The lattice model uniquely specifies reversal of time for the rank-twoDirac Hamiltonian, namely conjugation by τ1 K, for

(τ1 K)HD (K τ1) = HD.


Let the Bloch Hamiltonian for one spin-1/2 fermion be

H(k) := −2t σ0 cos k , −π ≤ k < +π,

(uniform nearest-neighbor hopping t > 0).Linearization of this dispersion about the two Fermi points ±kF with−π < kF < +π gives the rank two Dirac Hamiltonian

HD := −τ3 ⊗ σ0 i∂

∂x(~ ≡ 1, vF = 2t | sin kF| ≡ 1).

The lattice model uniquely specifies reversal of time for the rank-fourDirac Hamiltonian, namely conjugation by τ1 ⊗ σ2 K, for

(τ1 ⊗ σ2 K)HD (K τ1 ⊗ σ2) = HD.


Assume that you are given the rank-two Dirac Hamiltonian

HD := −τ3 i∂

∂x.

In how many ways can you define reversal of time?Reversal of time involves complex conjuation since we must undo the signchange of the time derivative in

i∂

∂tΨ(x , t) = HD Ψ(x , t)

under t 7→ −t . However, complex conjugation alone reverses the sign of themomentum operator

p := −i∂

∂xon the right-hand side of the Dirac equation. We can undo this change of signby conjugation with either τ1 or τ2,

HD =

{(τ1 K)HD (K τ1),

(τ2 K)HD (K τ2).

With no reference to a microscopic model, there is no unique way to definereversal of time (This is not so for Weyl Hamiltonian in 3D!).


Tutorial on Anderson localization:Schrödinger equation with a delta-function double well

Hdw = −12

d2

dx2 − v [δ(x + d/2) + δ(x − d/2)]

in units with ~ = m = 1.


Tutorial on Anderson localization:Schrödinger equation with short-range correlatedpotential disorder

In units with ~ = m = 1, consider the random Hamiltonian

H = −12

d2

dx2 + V (x),

with the vanishing meanV (x) = 0,

the second moment

V (x) V (y) = g2v e−|x−y |/ξdis ,

and all higher moments vanishing. Locality means that ξdis <∞.Typically, all states are exponentially localized for this one-dimensionalrandom Hamiltonian.


Tutorial on Anderson localization:Tight-binding model with “weak” short-rangecorrelated disorderStart from

(a) (b)

Break “weakly” translation invariance by adding uncorrelated on-sitepotentials µi and uncorrelated nearest-neighbor hopping ti = t eiφi ofvanishing means. You then get

H = −iτ3∂

∂x+ a0(x) τ0 + m1(x) τ1 + m2(x) τ2 + a1(x) τ3

with a0, m1, m2, and a1 uncorrelated beyond some length scale ξdis.C. Mudry (PSI) Interactions in Topological Matter 41 / 108

Tutorial on Anderson localization:The quasi-one-dimensional case

Define the quasi-d-dimensional Dirac Hamiltonian

H(x) = −i(α⊗ I) · ∂∂x

+ V(x), (1a)

where α and β are a set of matrices that anticommute pairwise andsquare to the unit rmin × rmin matrix, I is a unit N × N matrix, and

V(x) = m(x)β ⊗ I + · · · (1b)

with · · · representing all other masses, vector potentials, and scalarpotentials allowed by the AZ symmetry class.


For one-dimensional space, the stationary eigenvalue problem

H(x) Ψ(x ; ε) = εΨ(x ; ε) (2)

with the given “initial value” Ψ(y ; ε) is solved through the transfermatrix

Ψ(x ; ε) =M(x |y ; ε) Ψ(y ; ε) (3a)

where

M(x |y ; ε) = Px ′ exp

x∫y

dx ′ i(α⊗ I) [ε− V(x ′)]

. (3b)

The symbol Px ′ represents path ordering. The limit N →∞ with allentries of V independently and identically distributed (iid) up to the AZsymmetry constraints, (averaging over the disorder is denoted by anoverline)

Vij(x) ∝ vij ,[Vij(x)− vij

] [Vkl(y)− vkl

]∝ g2 e−|x−y |/ξdis , (4)

for i , j , k , l = 1, · · · , rmin N defines the thick quantum wire limit.C. Mudry (PSI) Interactions in Topological Matter 43 / 108

The consequences of Eq. (3) are the following.

First, the local symmetries defining the symmetry classes A, AII, andAI obeyed by ε− V(x ′) carry through to the transfer matrix at anysingle-particle energy ε.

The local unitary spectral symmetries defining the symmetry classesAIII, CII, and BDI and the local anti-unitary spectral symmetriesdefining the symmetry classes D, DIII, C, and CI carry through to thetransfer matrix at the single-particle energy ε = 0.


Second, the diagonal matrix entering the polar decomposition of thetransfer matrix at the band center ε = 0 is related to the non-compactsymmetric spaces from the columnM in Table


DIII Y N 2 0 2 D DIII +2/3 4√




Third, the composition law obeyed by the transfer matrix that encodesenlarging the length of a disordered wire coupled to perfect leads ismatrix multiplication. It is then possible to derive a Fokker-Planckequation for the joint probability obeyed by the radial coordinates onthe non-compact symmetric spaces from the columnM in the Table asthe length L of of a disordered wire coupled to perfect leads isincreased.


In this way, the moments of the dimensionless Landauer conductanceg in the columns δg and −ln g can be computed.An infinitesimal increase in the length of the disordered region for oneof the ten symmetry classes induces an infinitesimal Brownian motionof the Lyapunov exponents that is solely controlled by the multiplicitiesof the ordinary, long, and short roots of the corresponding classicalsemi-simple Lie algebra under suitable assumptions on the disorder(locality, weakness, and isotropy between all channels).


DIII Y N 2 0 2 D DIII +2/3 4√




When the transfer matrix describes the stability of the metallic phase inthe thick quantum wire limit of non-interacting fermions perturbed bystatic one-body random potentials with local correlations and ofvanishing means in the bulk of a quasi-one-dimensional lattice model,the multiplicities of the short root entering the Brownian motion of theLyapunov exponents always vanish.


DIII Y N 2 0 2 D DIII +2/3 4√




However, when the transfer matrix describes thequasi-one-dimensional boundary of a two-dimensional topologicalband insulator moderately perturbed by static one-body randompotentials with local correlations, the multiplicities of the short roots isnonvanishing in the Brownian motion of the Lyapunov exponents in thefive AZ symmetry classes A, AII, D, DIII, and C.

Correspondingly, the conductance is of order one along the infinitelylong boundary, i.e., the insulating bulk supports extended edge states.

These extended edge states can be thought of as realizing aquasi-one-dimensional ballistic phase of quantum matter robust todisorder.



Introduction with tutorialsX

The tenfold way in quasi-one-dimensional space





Symmetry class A and r = 2:

The Dirac Hamiltonian

H = τ3 k + A0 τ0 + M1 τ1 + M2 τ2 + A1 τ3 (5a)

is said to belong to the AZ symmetry class A. The set

V Ad=1,r=2 := { β(θ) := τ1 cos θ + τ2 sin θ|0 ≤ θ < 2π} =: S1, (5b)

a circle, is the topological space of normalized Dirac massesassociated with the Dirac Hamiltonian. Note that V A

d=1,r=2 and U(1)are homeomorphic as topological spaces. Thus, they share the samehomotopy groups. On the other hand, V A

d=1,r=2 is not a group undermatrix multiplication, while U(1) is.


Symmetry class AII and r = 2:

If charge conservation holds and TRS is imposed through

H(k) = +τ2H∗(−k) τ2, (6a)

thenH(k) = τ3 k + τ0 A0. (6b)

No mass matrix is permissible if TRS squares to minus the identity.The topological space of normalized Dirac masses in the symmetryclass AII is the empty set

V AIId=1,r=2 = ∅. (6c)

Because of the fermion-doubling problem, the only way to realize (6b)as the low-energy and long wavelength limit of a lattice model withshort-range correlated disorder is on the boundary of atwo-dimensional topological insulator in the symmetry class AII.


Symmetry class AI and r = 2:If charge conservation holds and TRS is imposed through

H(k) = +τ1H∗(−k) τ1, (7a)

thenH(k) = τ3 k + τ2 M2 + τ1 M1 + τ0 A0. (7b)

The same mass matrix as in the symmetry class A is permissible ifTRS squares to the identity. The homeomorphy between the allowedmasses in the symmetry classes A and AI is accidental. It does nothold for larger representations of the Dirac matrix. The topologicalspace of normalized Dirac masses is

V AId=1,r=2 := {β(θ)|0 ≤ θ < 2π} =: S1. (7c)

The topological spaces V AId=1,r=2 and U(1) are homeomorphic. [This

homeomorphism is not a group homomorphism, for V Ad=1,r=2 is not a

group while U(1) is.] Consequently, they share the same homotopygroups.


Symmetry class AIII and r = 2:

If charge conservation holds together with the CHS

H(k) = −τ1H(k) τ1, (8a)

thenH(k) = τ3 k + τ3 A1 + τ2 M2. (8b)

There is a unique mass matrix. The topological space of normalizedDirac masses obtained by adding to the Dirac kinetic contribution amass matrix squaring to unity and obeying the CHS is

V AIIId=1,r=2 = {±τ2}. (8c)


Symmetry class CII and r = 2:

It is not possible to write down a 2× 2 Dirac equation in the symmetryclass CII. For example, imposing

H(k) = −τ1H(k) τ1, H(k) = +τ2H∗(−k) τ2, (9)

enforces the symmetry class DIII, for composing the CHS with the TRSdelivers a PHS that squares to the unity and not minus the unity. Inorder to implement the symmetry constraints of class CII, we need toconsider a 4× 4 Dirac equation.


Symmetry class BDI and r = 2:

If charge conservation holds together with

H(k) = −τ1H(k) τ1, H(k) = +τ1H∗(−k) τ1, (10a)

thenH(k) = τ3 k + τ2 M2. (10b)

There is a unique mass matrix. The topological space of normalizedDirac masses obtained by adding to the Dirac kinetic contribution amass matrix squaring to unity while preserving TRS and PHS (aproduct of TRS and CHS), both of which square to unity, is

V BDId=1,r=2 = {±τ2}. (10c)


Symmetry class D and r = 2:

If we impose PHS through

H(k) = −H∗(−k), (11a)

thenH(k) = τ3 k + τ2 M2. (11b)

There is a unique mass matrix. The topological space of normalizedDirac masses obtained by adding to the Dirac kinetic contribution amass matrix squaring to unity and preserving the PHS squaring tounity is

V Dd=1,r=2 = {±τ1}. (11c)


Symmetry class DIII and r = 2:If we impose PHS and TRS through

H(k) = −H∗(−k), H(k) = +τ2H∗(−k) τ2, (12a)

respectively, thenH(k) = τ3 k . (12b)

No mass matrix is permissible if TRS squares to minus the identity.The topological space of normalized Dirac masses in the symmetryclass DIII is the empty set

V DIIId=1,r=2 = ∅. (12c)

Because of the fermion-doubling problem,the only way to realize (12b)as the low-energy and long wavelength limit of a lattice model withshort-range correlated disorder is on the boundary of atwo-dimensional topological superconductor in the symmetry classDIII.


Symmetry class C and r = 2:

If we impose PHS through

H(k) = −τ2H∗(−k) τ2, (13a)

thenH(k) = τ3 A1 + τ2 M2 + τ1 M1. (13b)

PHS squaring to minus unity prohibits a kinetic energy in any DiracHamiltonian of rank 2 in the symmetry class C.


Symmetry class CI and r = 2:

If we impose PHS and TRS through

H(k) = −τ2H∗(−k) τ2, H(k) = +τ1H∗(−k) τ1, (14a)

respectively, thenH(k) = τ2 M2 + τ1 M1. (14b)

PHS squaring to minus unity prohibits a kinetic energy in the symmetryclass CI.


Symmetry class A and r = 4:The Dirac Hamiltonian in the symmetry class A is

H(k) :=τ3 ⊗ σ0 k + τ3 ⊗ σν A1,ν + τ2 ⊗ σν M2,ν

+ τ1 ⊗ σν M1,ν + τ0 ⊗ σν A0,ν .(15a)

The topological space of normalized Dirac masses is

V Ad=1,r=4 :=

{β =

(0 U

U† 0

)∣∣∣∣U ∈ U(2)

}. (15b)

As a topological space, it is thus homeomorphic toU(2) ' U(1)× SU(2) ' S1 × S3, an interpretation rendered plausible by theparameterization

V Ad=1,r=4 =

{M · X+N · Y |M2 = cos2 θ,N = tan θM

}= S1 × S3,

M :=(M2,0,M1,1,M1,2,M1,3

), N :=

(−M1,0,M2,1,M2,2,M2,3

),

X := (τ2 ⊗ σ0, τ1 ⊗ σ1, τ1 ⊗ σ2, τ1 ⊗ σ3) ,

Y := (−τ1 ⊗ σ0, τ2 ⊗ σ1, τ2 ⊗ σ2, τ2 ⊗ σ3) .

(15c)


Symmetry class AII and r = 4:If charge conservation holds together with TRS through

H(k) = +τ1 ⊗ σ2H∗(−k) τ1 ⊗ σ2, (16a)

then

H(k) = τ3 ⊗ σ0 k +∑

ν=1,2,3

τ3 ⊗ σν A1,ν + τ2 ⊗ σ0 M2,0

+ τ1 ⊗ σ0 M1,0 + τ0 ⊗ σ0 A0,0.

(16b)

Observe that by doubling the Dirac Hamiltonian (6b), we went from nomass matrix to two anticommuting mass matrices. The topologicalspace of normalized Dirac masses is

V AIId=1,r=4 :=

{β =

(0 U

U† 0

)∣∣∣∣U = +σ2 UTσ2 ∈ U(2)

}. (16c)


As a topological space, V AIId=1,r=4 can be shown to be homeomorphic to

U(2)/Sp(1) ' U(1)× SU(2)/SU(2) ' U(1), an interpretationrendered plausible by the parameterization

V AIId=1,r=4 =

{M · X |M2 = 1

}=: S1,

M :=(M2,0,M1,0

), X := (τ2 ⊗ σ0, τ1 ⊗ σ0) .


Symmetry class AI and r = 4:If charge conservation holds together with TRS through

H(k) = +τ1 ⊗ σ0H∗(−k) τ1 ⊗ σ0, (17a)

then

H(k) = τ3 ⊗ σ0 k + τ3 ⊗ σ2 A1,2 +∑

ν=0,1,3

(τ2 ⊗ σν M2,ν

+ τ1 ⊗ σν M1,ν + τ0 ⊗ σν A0,ν).

(17b)

There are six mass matrices of rank r = 4 in the 1D symmetry class AIthat can be arranged into the three pairs (M1,ν ,M2,ν) with ν = 0,1,3 ofanticommuting masses. The topological space of normalized Diracmasses is

V AId=1,r=4 :=

{β =

(0 U

U† 0

)∣∣∣∣U = +UT ∈ U(2)

}. (17c)


As a topological space, V AId=1,r=4 can be shown to be homeomorphic to

U(2)/O(2) ' U(1)/{±1} × SU(2)/U(1) ' S1 × S2, an interpretationrendered plausible by the parameterization

V AId=1,r=4 =

{M · X +N · Y |M2 = cos2 θ,N = tan θM

}=: S1 × S2,

M :=(M2,0,M1,1,M1,3

), N :=

(−M1,0,M2,1,M2,3

),

X := (τ2 ⊗ σ0, τ1 ⊗ σ1, τ1 ⊗ σ3) , Y := (−τ1 ⊗ σ0, τ2 ⊗ σ1, τ2 ⊗ σ3) .


Symmetry class AIII and r = 4:If charge conservation holds together with the CHS

H(k) = −τ1 ⊗ σ0H(k) τ1 ⊗ σ0, (18a)

then

H(k) = τ3 ⊗ σ0 k +∑

ν=0,1,2,3

(τ3 ⊗ σν A1,ν + τ2 ⊗ σν M2,ν

). (18b)

The Dirac mass matrix τ2 ⊗ σ0 M2,0 that descends from Eq. (8b)commutes with the triplet of anticommuting mass matricesτ2 ⊗ σ1 M2,1, τ2 ⊗ σ2 M2,2, and τ2 ⊗ σ3 M2,3. The topological space ofnormalized Dirac masses is

V AIIId=1,r=4 :=

{β = τ2 ⊗ A

∣∣∣∣∣A := U Im,n U†,

m,n = 0,1,2, m + n = 2, U ∈ U(2),

Im,n := diag(

m−times︷︸︸︷−1, . . . ,−1,

n−times︷︸︸︷+1, . . . ,+1)

}.


As a topological space, it is thus homeomorphic toU(2)/[U(2)× U(0)] ∪ U(2)/[U(1)× U(1)] ∪ U(2)/[U(0)× U(2)], as isalso apparent from the parameterization

V AIIId=1,r=4 = {±τ2 ⊗ σ0} ∪

{M · X |M2 = 1

},

M :=(M2,1,M2,2,M2,3

),

X := (τ2 ⊗ σ1, τ2 ⊗ σ2, τ2 ⊗ σ3) ,

(recall that S2 ' SU(2)/U(1) so that U(2)/[U(1)× U(1)] ' S2).


Symmetry class CII and r = 4:If charge conservation holds together with CHS and TRS

H(k) = −τ1 ⊗ σ0H(k) τ1 ⊗ σ0, (19a)H(k) = +τ1 ⊗ σ2H∗(−k) τ1 ⊗ σ2, (19b)

respectively, then

H(k) = τ3 ⊗ σ0 k +∑

ν=1,2,3

τ3 ⊗ σν A1,ν + τ2 ⊗ σ0 M2,0. (19c)

There is a unique mass matrix, as was the case in Eqs. (8b) and (10b).The topological space of normalized Dirac masses is

V CIId=1,r=4 :=

{β ∈ V AIII

d=1,r=4

∣∣∣β = (τ1 ⊗ σ2)β∗ (τ1 ⊗ σ2)}. (19d)

As a topological space, V CIId=1,r=4 can be shown to be homeomorphic to

Sp(1)/Sp(1)× Sp(0) ∪ Sp(1)/Sp(0)× Sp(1), an interpretationrendered plausible by the parameterization

V CIId=1,r=4 = {±τ2 ⊗ σ0}. (19e)


Symmetry class BDI and r = 4:

If charge conservation holds together together with CHS and TRS

H(k) = −τ1 ⊗ σ0H(k) τ1 ⊗ σ0, (20a)H(k) = +τ1 ⊗ σ0H∗(−k) τ1 ⊗ σ0, (20b)

respectively, then

H(k) = τ3 ⊗ σ0 k + τ3 ⊗ σ2 A1,2 +∑

ν=0,1,3

τ2 ⊗ σν M2,ν . (20c)

The Dirac mass matrix τ2 ⊗ σ0 M2,0 that descends from Eq. (10b)commutes with the pair of anticommuting mass matrices τ2 ⊗ σ1 M2,1and τ2 ⊗ σ3 M2,3. The topological space of normalized Dirac masses is

V BDId=1,r=4 :=

{β ∈ V AIII

d=1,r=4

∣∣∣β = (τ1 ⊗ σ0)β∗ (τ1 ⊗ σ0)}. (20d)


As a topological space, V BDId=1,r=4 can be shown to be homeomorphic to

O(2)/[O(2)×O(0)] ∪O(2)/[O(1)×O(1)] ∪O(2)/[O(0)×O(2)], aninterpretation rendered plausible from the parameterization

V BDId=1,r=4 = {±τ2 ⊗ σ0} ∪

{M · X |M2 = 1

},

M :=(M2,1,M2,3

),

X := (τ2 ⊗ σ1, τ2 ⊗ σ3) ,

(recall that S1 ' O(2)/[O(1)×O(1)]).


Symmetry class D and r = 4:If we impose PHS through

H(k) = −H∗(−k), (21a)

then

H(k) = τ3 ⊗ σ0 k + τ3 ⊗ σ2 A1,2 +∑

ν=0,1,3

τ2 ⊗ σν M2,ν

+ τ1 ⊗ σ2 M1,2 + τ0 ⊗ σ2 A0,2.

(21b)

There are four Dirac mass matrices. None commutes with all remaining ones.However, each of them is antisymmetric and so is their sum. The topologicalspace of normalized Dirac masses is

V Dd=1,r=4 =

{M · X |M2 = 1

}∪{

N · Y |N2 = 1},

M :=(M2,1,M2,3

), N :=

(M2,0,M1,2

),

X := (τ2 ⊗ σ1, τ2 ⊗ σ3) , Y := (τ2 ⊗ σ0, τ1 ⊗ σ2) .

(22)

As a topological space, V Dd=1,r=4 is homeomorphic to O(2), as

V Dd=1,r=4 ' S1 ∪ S1 ' U(1)× Z2 ' O(2).


Symmetry class DIII and r = 4:

If we impose PHS and TRS through

H(k) = −H∗(−k), H(k) = +τ2 ⊗ σ0H∗(−k) τ2 ⊗ σ0, (23a)

thenH(k) = τ3 ⊗ σ0 k + τ3 ⊗ σ2 A1,2 + τ1 ⊗ σ2 M1,2. (23b)

Observe that there is only one Dirac mass matrix [there was none inEq. (12b)]. Moreover, this Dirac mass matrix is Hermitian andantisymmetric. The topological space of normalized Dirac masses is

V DIIId=1,r=4 = {±τ1 ⊗ σ2} . (23c)

As a topological space, V DIIId=1,r=4 is homeomorphic to O(2)/U(1).


Symmetry class C and r = 4:If we impose PHS through

H(k) = −τ0 ⊗ σ2H∗(−k) τ0 ⊗ σ2, (24a)

then

H(k) = τ3 ⊗ σ0 k +∑

ν=1,2,3

τ3 ⊗ σν A1,ν + τ2 ⊗ σ0 M2,0

+∑

ν=1,2,3

τ1 ⊗ σν M1,ν +∑

ν=1,2,3

τ0 ⊗ σν A0,ν .(24b)

There are four mass matrices that anticommute pairwise. The topologicalspace of normalized Dirac masses is

V Cd=1,r=4 =

{M · X |M2 = 1

}=: S3, M :=

(M2,0,M1,1,M1,2,M1,3

),

X := (τ2 ⊗ σ0, τ1 ⊗ σ1, τ1 ⊗ σ2, τ1 ⊗ σ3) .(24c)

As a topological space, V Cd=1,r=4 is homeomorphic to Sp(1) since we have

Sp(1) ' SU(2) ' S3.


Symmetry class CI and r = 4:If we impose PHS and TRS through

H(k) = −τ0 ⊗ σ2H∗(−k) τ0 ⊗ σ2, H(k) = +τ1 ⊗ σ0H∗(−k) τ1 ⊗ σ0, (25a)

respectively, then

H(k) = τ3 ⊗ σ0 k + τ3 ⊗ σ2 A1,2 + τ2 ⊗ σ0 M2,0

+∑ν=1,3

(τ1 ⊗ σν M1,ν + τ0 ⊗ σν A0,ν

). (25b)

There are three mass matrices that anticommute pairwise. The topologicalspace of normalized Dirac masses is

V CId=1,r=4 =

{M i · X i |M2

i = 1}

=: S2, M :=(M2,0,M1,1,M1,3

),

X := (τ2 ⊗ σ0, τ1 ⊗ σ1, τ1 ⊗ σ3) .(26)

As a topological space, V CId=1,r=4 is homeomorphic to Sp(1)/U(1) since we

have the homeomorphism Sp(1)/U(1) ' SU(2)/U(1) ' S2.


Classifying spaces

Label Classifying space VC0 ∪N

n=0{

U(N)/[U(n)× U(N − n)

]}C1 U(N)

R0 ∪Nn=0{

O(N)/[O(n)×O(N − n)

]}R1 O(N)R2 O(2N)/U(N)R3 U(2N)/Sp(N)R4 ∪N

n=0{

Sp(N)/[Sp(n)× Sp(N − n)

]}R5 Sp(N)R6 Sp(N)/U(N)R7 U(N)/O(N)


Path connectedness of the normalized Dirac masses

Case (a): π0(V ) = {0} Trivial phase

Case (b): π0(V ) = Z

..... .....

..... .....

NEven

Odd N

ν=+1ν=0ν=−1

ν=−1/2 ν=+1/2

Case (c): π0(V ) = Z2 ν=0 ν=1


A

A

A

B

BB

B B

A

B

A

B


AZ symmetry class rmin Vd=1,r π0(Vd=1,r ) Phase diagram from the Figure below Cut at m = 0A 2 C1 0 (a) insulating

AIII 2 C0 Z (b) even-oddAI 2 R7 0 (a) insulating

BDI 2 R0 Z (b) even-oddD 2 R1 Z2 (c) critical

DIII 4 R2 Z,2 (c) criticalAII 4 R3 0 (a) insulatingCII 4 R4 Z (b) even-oddC 4 R5 0 (a) insulatingCI 4 R6 0 (a) insulating

Insulator Insulator

ν=0 ν=1

(b) D, DIII

0

Insulator

(c) A, AI, AII, C, CI(a) AIII, BDI, CII

N odd N even

ν=−1/2 ν=1/2

ν=1ν=−1 ν=0

0 0

ν=N/2ν=−N/2

... ...

ν=N/2ν=−N/2

... ...




The tenfold way in quasi-one-dimensional space X






Universal data: Kij = Kji ∈ Z, with det K 6= 0 and qi ∈ Z

where (−1)Kii = (−1)

qi ,.

Non-universal data: Vij = Vji ∈ Z, positive definite matrix.

H :=

L∫0

dx

[1

4πVij(Dx ui

) (Dx uj

)

+ A0

( qi

2πK−1

ij

(Dx uj

))](t, x),

Dx ui (t, x) :=(∂x ui + qi A1

)(t, x),[

ui (t, x), uj (t, y)]

= iπ[Kij sgn(x − y) + Lij

],

Lij = −Lji =

0, if i = j,

sgn(i − j)(

Kij + qi qj

), otherwise,

ui (t, x + L) = ui (t, x) + 2πni , ni ∈ Z.

Chiral equations of motions: 0 = δik D0 uk + Kij Vjk D1 uk

Anomalous continuity equation: ∂µ J µ = σH∂ A1∂t where

J0(t, x) =1

2πqi K−1

ij

(D1 uj

)(t, x),

J1(t, x) =1

2πqi Vij

(D1 uj

)(t, x) + σH A0(t, x)

with σH ≡1

2π

(qi K−1

ij qj

).

B

x

y

z

(a) (b)


ApplicationThe conserved charge

Q =

∫R

dx(

ˆψ γ0 ψ)

(t , x)→ ε01

2π

[φ(t , x = +∞)− φ(t , x = −∞)

](27)

for the static profile ϕ(x) is approximately given by

Q ≈ ε01

2π[ϕ(x = +∞)− ϕ(x = −∞)] . (28)

On the other hand, the number of electrons per periode T = 2π/ω thatflows across a point x

I =

T∫0

dt(

ˆψ γ1 ψ)

(t , x)→ ε10

2π

[φ(T , x)− φ(0, x)

](29)

for the uniform profile ϕ(t) = ω t is approximately given

I ≈ ε10

2πω T = ε10. (30)


Tutorial on the anomalous continuity equationDefine the quantum Hamiltonian (in units with the electric charge e,the speed of light c, and ~ set to one)

H =

L∫0

dx[

14π

Vij (Dx ui)(

Dx uj

)+ A0

( qi2π

K−1ij

(Dx uj

))](t , x),

Dx ui(t , x) := (∂x ui + qi A1) (t , x).

(31)

The indices i , j = 1, · · · ,N label the bosonic modes. Summation isimplied for repeated indices. The N real-valued quantum fields ui(t , x)obey the equal-time commutation relations[

ui(t , x), uj(t , y)]

= iπ[Kij sgn(x − y) + Lij

](32)

for any pair i , j = 1, · · · ,N. The function sgn(x) = −sgn(−x) gives thesign of the real variable x and will be assumed to be periodic withperiodicity L.


The N × N matrix K is symmetric, invertible, and integer valued. Giventhe pair i , j = 1, · · · ,N, any of its matrix elements thus obey

Kij = Kji ∈ Z, K−1ij = K−1

ji ∈ Q. (33)

The N × N matrix L is anti-symmetric

Lij = −Lji =

0, if i = j ,

sgn(i − j)(

Kij + qi qj

), otherwise,

(34)

for i , j = 1, · · · ,N. The sign function sgn(i) of any integer i is here notmade periodic and taken to vanish at the origin of Z.The N × N matrix V is symmetric and positive definite

Vij = Vji ∈ R, vi Vij vj > 0, i , j = 1, · · · ,N, (35)

for any nonvanishing vector v = (vi ) ∈ RN . The charges qi are integervalued and satisfy

(−1)Kii = (−1)qi , i = 1, · · · ,N. (36)


The external scalar gauge potential A0(t , x) and vector gauge potentialA1(t , x) are real-valued functions of time t and space x coordinates.They are also chosen to be periodic under x 7→ x + L.

Finally, we shall impose the boundary conditions

ui(t , x + L) = ui(t , x) + 2πni , ni ∈ Z, (37)

and(∂x ui) (t , x + L) = (∂x ui) (t , x), (38)

for any i = 1, · · · ,N.

First important result: The equations of motion

0 = δik D0 uk + Kij Vjk D1 uk , i = 1, · · · ,N, (39)

are chiral.


Proof of anomalous continuity equationSecond important result:The anomalous continuity equation

∂µJµ = +σH∂ A1∂t

(40)

holds.

Proof.With the help of[

Dx ui(t , x),Dy uj(t , y)]

= −2πi Kij δ′(x − y) (41)

for i , j = 1, · · · ,N, one verifies that the total derivative of J0(t , x) is

∂ J0

∂t= − i

[J0, H

]+ σH

∂ A1∂t

= − ∂ J1

∂x+ σH

∂ A1∂t

. (42)


Proof.Alternatively, introduce the density

ρ(t , x) =1

2πqi K−1

ij

(∂x uj

)(t , x) (43a)

and the current density

j(t , x) =1

2πqi Vij

(∂x uj

)(t , x). (43b)

First, verify that taking the divergence over Jµ gives

∂µ Jµ ≡ ∂t J0 + ∂x J1

= ∂t ρ+ σH ∂t A1︸︷︷︸+∂x j +1

2π(qi Vij qj

)∂x A1 + σH ∂x A0. (44)

Second, verify with the help of the chiral equations of motion that

∂t ρ+ ∂x j = − 12π

(qi Vij qj

)∂x A1 − σH ∂x A0. (45)


Tutorial on quasi-particle (anyons) and Fermi-BoseexcitationsA first application of the Baker-Campbell-Hausdorff formula to any pairof quasi-particle vertex operator at equal time t but two distinct spacecoordinates x 6= y gives

Ψ†q-p,i(t , x) Ψ†q-p,j(t , y) = e−iπΘq-pij Ψ†q-p,j(t , y) Ψ†q-p,i(t , x), (46a)

where

Θq-pij = K−1

ji sgn(x − y) +(

K−1ik K−1

jl Kkl + qk K−1ik K−1

jl ql

)sgn(k − l).

(46b)Here and below, it is understood that

sgn(k − l) = 0 (47)

when k = l = 1, · · · ,N. Hence, the quasi-particle vertex operatorsobey neither bosonic nor fermionic statistics since K−1

ij ∈ Q.


The same exercise applied to the Fermi-Bose vertex operators yields

Ψ†f-b,i(t , x) Ψ†f-b,j(t , y) =

(−1)Kii Ψ†f-b,i(t , y) Ψ†f-b,i(t , x), if i = j ,

(−1)qi qj Ψ†f-b,j(t , y) Ψ†f-b,i(t , x), if i 6= j ,(48)

when x 6= y . The self statistics of the Fermi-Bose vertex operators iscarried by the diagonal matrix elements Kii ∈ Z. The mutual statisticsof any pair of Fermi-Bose vertex operators labeled by i 6= j is carried bythe product qi qj ∈ Z of the intger-valued charges qi and qj . Had wenot assumed that Kij with i 6= j are integers, the mutual statistics wouldnot be Fermi-Bose because of the non-local term Kijsgn (x − y).


A third application of the Baker-Campbell-Hausdorff formula allows todetermine the boundary conditions

Ψ†q-p,i(t , x + L) = Ψ†q-p,i(t , x) e−2πi K−1ij Ni e−πi K−1

ii (49)

andΨ†f-b,i(t , x + L) = Ψ†f-b,i(t , x) e−2πi Ni e−πi Kii (50)

obeyed by the quasi-particle and Fermi-Bose vertex operators,respectively.


Tutorial on the Abelian bosonization rules

Table: Abelian bosonization rules in two-dimensional Minkowski space. The conventions of relevance to the scalar mass

ˆψ ψ and the pseudo-scalar mass ˆψ γ5 ψ are ˆψ = ψ† γ0 with ψ† = (ψ†−, ψ

†+), whereby γ0 = τ1 and γ1 = iτ2 so that

γ5 = −γ5 = −γ0 γ1 = τ3.

Fermions Bosons

Kinetic energy ˆψ iγµ ∂µ ψ1

8π (∂µ φ)(∂µ φ)

Current ˆψ γµ ψ 12π ε

µν ∂ν φ

Chiral currents 2 ψ†∓ ψ∓ ± 12π ∂x u∓

Right and left movers ψ†∓

√1

4π ae∓iu∓

Backward scattering ψ†− ψ+

14π a

e−iφ

Cooper pairing ψ†− ψ†+

14π a

e−iθ

Scalar mass ψ†− ψ+ + ψ

†+ ψ−

12π a

cos φ

Pseudo-scalar mass ψ†− ψ+ − ψ

†+ ψ−

−i2π a

sin φ





Fractionalization from Abelian bosonization X




Stability analysis for the edge theory in the symmetryclass AII

The edge of a two-dimensional insulator in the symmetry class AII is described by

H := H0 + Hint, H0 :=

L∫0

dx1

4π∂x ΦT V ∂x Φ, Hint := −

L∫0

dx∑T∈L

hT (x) : cos(

T TK Φ(x) + αT (x))

: . (51a)

The real functions hT (x) ≥ 0 and 0 ≤ αT (x) ≤ 2π encode information about the disorder along the edge when positiondependent. The set

L :={

T ∈ Z2N∣∣∣T TQ = 0

}, (51b)

encodes all the possible charge neutral tunneling processes (i.e., those that just rearrange charge among the branches).The theory is quantized according to the equal-time commutators

[Φi (t, x), Φj (t, x′)

]= −iπ

(K−1

ij sgn(x − x′) + Θij

), (51c)

where K is a 2N × 2N symmetric and invertible matrix with integer-valued matrix elements, and the Θ matrix accounts for Kleinfactors that ensure that charged excitations in the theory (vertex operators) satisfy the proper commutation relations.


Tutorial on quasi-particle/Fermi-Bose vertex operatorsThe universal data are (K ,Q). The non-universal data are

(V , hT (x), αT (x)

). The boundary conditions

Kij Φj (t, x + L) = Kij Φj (t, x) + 2πni (52)

with ni ∈ Z for all i = 1, . . . , 2N are imposed. Together with the condition that the tunneling vectors T are restricted to haveinteger-valued components, this ensures that the Hamiltonian H is single-valued.The chiral nature of the bosonic quantum fields arises from demanding that the equal-time commutator

[Φi (t, x), Φj (t, x′)

]= −iπ

(K−1

ij sgn(x − x′) + Θij

)(53)

holds for any pair i, j = 1, . . . , 2N. Here,Θij := K−1

ik Lkl K−1lj (54)

and the antisymmetric 2N × 2N matrix L is defined by

Lij = sgn(i − j)(

Kij + Qi Qj

), (55)

where sgn(0) = 0 is understood. It then follows that the quadratic theory in Eq. (51) is endowed with chiral equations of motion.Finally, we need to impose the compatibility conditions

(−1)Kii = (−1)

Qi , i = 1, . . . , 2N, (56)

in order to construct local excitations with well-defined statistics.


Define for any i = 1, . . . , 2N the pair of normal-ordered vertex operators

Ψ†q-p,i (t, x) := : e

−iδij Φj (t,x):, Ψ

†q-p,T (t, x) := : e−iT T Φ(t,x) :,

Ψ†f-b,i (t, x) := : e

−iKij Φj (t,x):, Ψ

†f-b,T (t, x) := : e−iT T K Φ(t,x) :,

(57)

respectively. For any i = 1, . . . , 2N, the quasiparticle vertex operator Ψ†q-p,T (t, x) is multi valued under the transformation (64a)

provided |det K | > 1 in contrast to the Fermi-Bose vertex operator Ψ†f-b,T (t, x) which is always single valued under the

transformation (64a). For any pair i, j = 1, · · · ,N, the equal-time commutator (53) delivers the identities

[Ni , Ψ

†q-p,j (t, x)

]= δij Ψ

†q-p,j (t, x),

[Ni , Ψ

†f-b,j (t, x)

]= Kij Ψ

†f-b,j (t, x), (58)[

Ci , Ψ†q-p,j (t, x)

]= K−1

ij Ψ†q-p,j (t, x),

[Ci , Ψ

†f-b,j (t, x)

]= δij Ψ

†f-b,j (t, x), (59)

respectively. The quasiparticle vertex operator Ψ†q-p,i (t, x) and the Fermi-Bose vertex operator Ψ

†f-b,i (t, x) are the eigenstates of

the conserved topological number and of the conserved number operators

Ni :=1

2πKij

L∫0

dx(∂x Φj

)(t, x) =

1

2πKij

[Φj (t, L)− Φj (t, 0)

], Ci :=

1

2π

[Φi (t, L)− Φi (t, 0)

], (60)

respectively.


The permutation statistics obeyed by any pair i, j = 1, . . . , 2N of quasiparticle and Fermi-Bose operators are

Ψ†q-p,i (t, x) Ψ

†q-p,j (t, x′) = Ψ

†q-p,j (t, x′) Ψ

†q-p,i (t, x) e

−iπ[

K−1ij sgn(x−x′)+Θij

],

Ψ†f-b,i (t, x) Ψ

†f-b,j (t, x′) = Ψ

†f-b,j (t, x′) Ψ

†f-b,i (t, x) e

−iπ[Kij sgn(x−x′)+Lij

],

(61a)

when x 6= x′, respectively. For any x 6= x′, demanding that the 2N × 2N matrix K and the 2N-component charge vector Q areinteger-valued together with the compatibility condition (56) is required to obtain local excitations carrying the Fermi-Bosepermutation statistics

Ψ†f-b,i (t, x) Ψ

†f-b,j (t, x′) = (−)1

Qi Qj Ψ†f-b,j (t, x′) Ψ

†f-b,i (t, x). (61b)

The charge vector Q enters explicitly the theory after coupling the 2N chiral scalar fields to an external vector gauge potential withthe components A0 and A1 through the minimal coupling and the addition of the contribution

∂x Φi → Dx Φi := ∂x Φi + Qi A1, Hcurrent :=

L∫0

dx1

2πA0

(QT Dx Φ

), (62)

respectively. The theory is then invariant under the pure U(1) electro-magnetic gauge transformation. The total charge operator isthen

Q := Qi Ci . (63)

It follows that the charge associated with the quasiparticle operator Ψ†q-p,i and with the Fermi-Bose operator Ψ

†f-b,i is given by

K−1ij Qj and Qi , respectively.


By assumption, the integer-valued 2N × 2N matrix K is symmetric andinvertible. Consequently, its inverse K−1 is also symmetric, but its matrixelements are rational numbers whenever |det K | > 1. Observe that themodel (51) is invariant under the transformation

Φ(t , x)→ Φ(t , x) + 2π T?, T? ∈ R2N , T T K T? ∈ Z, QTT? = 0, (64a)

for all tunneling vectors T ∈ L. The quantum Hamiltonian (51) thuspossesses an emergent global[

U(1)]2N

=[U(1)× U(1)

]N (64b)

symmetry compared to the microscopic model.

The set of all rational-valued vectors T? that satisfy conditions (64a) is thelattice L? dual to the lattice L. When |det K | > 1, the lattice L is a sublattice ofthe dual lattice L? that is generated by the quasiparticles carrying a unittopological charge. The ground state of Hamiltonian (51) with the periodicboundary conditions corresponding to the geometry of a torus is thendegenerate with the degeneracy |det K | > 1, which is nothing but the volumeof the unit cell of the lattice L in units of the unit cell of the dual lattice L∗.


Tutorial on connection to Chern-Simon theories

The universal data on the edge (K ,Q) are in one to one correspondence with atopological field theory in the (2 + 1)-dimensional field theory of the bulk (bulk-edgecorrespondence) given by

S = S0 + Se + Ss, (65a)

whereS0 = −

∫dt d2x εµνρ

14π

Kij aiµ ∂ν aj

ρ, (65b)

Se =

∫dt d2x εµνρ

e2π

Qi Aµ∂ν aiρ, (65c)

andSs =

∫dt d2x εµνρ

s2π

Si Bµ∂ν aiρ. (65d)

The background fields Aµ and Bµ dictate the couplings to conserved charge and spincurrents, owing to the emergent[

U(1)]2N

=[U(1)× U(1)

]N (65e)

local gauge symmetry when the space time is a closed manifold.


Tutorial on different usages of “Topology”Topology in physics is often, not always, signaled by the fact thatboundary conditions play a role in the thermodynamic limit:

j = N � 1

j = N � 1T (0)

a) b)

j = 1

j = 1

“Topology” comes in different flavors:The IQHE and FQHE both share a quantized response function,the Hall conductivity. This quantization has a topological origin.The FQHE has another topological attribute, absent for the IQHE,the (topological) ground state degeneracy.


Short-range entanglement (SRE) versus long-rangeentanglement (LRE)

The SRE paradigm for which many-body interactions are not toostrong: IQHE is an example of a Chern insulator, topological bandinsulator, topological insulator, or short-range entangled (SRE) phase.

Topological attributes of STP phases can be understood solely interms of anomalous transport on the spatial interface between distinctSRE phases.

The LRE paradigm for which many-body interactions areessential: FQHE is an example of a topologically ordered phase,fractional topological insulator, or long-range entangled phase (LRE)phase.

Topological attributes descend from intrinsic bulk properties.


There is a sense in which LRE phases are morerobust than SRE phases against a weak breaking oftheir defining symmetry

The topological attributes of LRE phases are not confined to theboundary in space between two distinct topological realizations ofthese phases, as they are for SRE phases. They also characterizeintrinsic bulk properties.

Hence, whereas gapless edge states are gapped by any breaking ofthe defining symmetry, topological bulk properties are robust to a weakbreaking of the defining symmetry as long as the characteristic energyscale for this symmetry breaking is small compared to the bulk gap inthe LRE phase.






Stability analysis for the edge theory in the symmetry class AII X



Construction of two-dimensional topological phasesfrom coupled wires

Yesterday

Today

j = N � 1

j = N � 1T (0)

a) b)

j = 1

j = 1


2

TABLE I. (Color online) Realization of a two-dimensional array of quantum wires in each symmetry class of the tenfold way.

Θ2 Π2 C2 Short-range entangled (SRE) topological phase Long-range entangled (LRE) topological phase

A 0 0 0 Z ��

��

AIII 0 0 + NONE

AII − 0 0 Z2��

��

��

��

��

DIII − + + Z2

��

��

��

��

��

D 0 + 0 Z ��

��

��

��

BDI + + + NONE

AI + 0 0 NONE

CI + − + NONE

C 0 − 0 Z��

��

��

��

��

��

CII − − + NONE


DefinitionsWe are after an interacting theory for chiral fermions

Ψ†(x) ≡(ψ†1(x) · · · ψ†MN(x)

), Ψ(x) ≡

ψ1(x)...

ψMN(x)

,

{ψa(x), ψ†a′(x

′)}

= δa,a′ δ(x − x ′), a,a′ = 1, · · · ,MN ≡ N .

We trade the chiral fermions for chiral bosons (Abelian bosonization)

Kaa′ = δii ′ Kγγ′ , Kaa′ = δii ′ Kγγ′ , γ, γ′ = 1, · · · ,M, i , i ′ = 1, · · · ,N,ψa(x) ≡ : exp

(+iKaa′ φa′(x)

),[

φa(x), φa′(x′)]

= −iπ(K−1

aa′ sgn(x − x ′) +K−1ab LbcK−1

ca′

),

Laa′ = sgn(a− a′) (Kaa′ + 1) .


The many-body Hamiltonian H for the MN interacting chiral fermions is

H = HV + H{T }, HV =

∫dx(∂x ΦT

)(x) V

(∂x Φ

)(x),

with

H{T } =

∫dx∑T

hT (x)

2

(e+iαT (x)

MN∏a=1

ψTaa (x) + H.c.

)

=

∫dx∑T

hT (x) cos(T TK Φ(x) + αT (x)

).

We demand that

V = (Vaa′) ≡(V(i,γ)(i′,γ′)

)= IN ⊗ (Vγγ′), T ≡ (Ta) ∈ ZN .

MN∑a=1

Ta =

0 mod 2, for D, DIII, C, and CI,

0, otherwise,

hT (x) = +hT (x) =≥ 0, αT (x) = α∗T (x) = −α−T (x) ∈ [0, 2π[.

The integer

q =MN∑a=1

|Ta|2

dictates that T encodes a q-body interaction in the fermion representation.C. Mudry (PSI) Interactions in Topological Matter 106 / 108

The Hamiltonian is TRS ifΘ H Θ−1 = +H. (66a)

This condition is met if

PΘ V P−1Θ = +V , (66b)

PΘ K P−1Θ = −K , (66c)

hT (x) = h−PΘT(x), (66d)

αT (x) = α−PΘT(x)− π T T PΘ IΘ. (66e)

The Hamiltonian is PHS ifΠ H Π−1 = +H. (67a)

This condition is met if

PΠ V P−1Π = +V , (67b)

PΠ K P−1Π = +K , (67c)

hT (x) = h+PΠT(x), (67d)

αT (x) = αPΠT(x) + π T T PΠ IΠ. (67e)


The many-body Hamiltonian H = HV + H{T } is to be chosen so that (i)it belongs to any one of the ten symmetry classes from the tenfold wayand (ii) all excitations in the bulk are gapped by a specific choice of thetunneling vectors {T } entering H{T }. The energy scales in H{T } are

assumed sufficiently large compared to those in HV so that it is HV thatmay be thought of as a perturbation of H{T } and not the converse.

We then apply the Haldane criterion

T TKT ′ = 0

for the choice we made for {T } to decide if there remain gapless edgestates when open boundary conditions are imposed.


Interactions in Topological Matter · Interactions in Topological Matter Christopher Mudry1 1Paul...

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