Many-Body Localization - DIAS
Transcript of Many-Body Localization - DIAS
Many-Body Localization
Wojciech De Roeck, Leuven
Francois Huveneers
Thimothée Thiery
DavidLuitz
Markus Müller
LuisColmenarez
Quantum spin chains I
● Hamiltonian
● Time evolution
● is conserved: restrict to sector
Hilbert space
Quantum spin chains I
● Hamiltonian
Hilbert space
Free fermions: Interaction:
Fermion number operator:
Quantum spin chains II
● Eigenstates
● Equilibrium ensemble
● Stay away from spectral edges: extensive entropy!
(microcanonical shell)
bulk energies
Thermalizing systems● Extreme non-equilibrium initial state
● Ergodic average
● Definition: System is thermalizing iff. (for large L)
Upshot: Thermalizing systems have transport over long distances
ETH: Eigenstate Thermalization Hypothesis (Deutsch, Srednicki, 91)
● Definition: System is thermalizing iff. (for large L)
● ETH: holds for all bulk eigenstates
● Thermalization ETH (if initial state has definite energy density)
● Sketch proof:
(Limit is reached at hence non-physical times)
● No proofs of ETH, but strong numerical evidence (Rigol et al 2008 -…)
ETH: Eigenstate Thermalization Hypothesis (Deutsch, Srednicki, 91)
● ETH: I)
II)
● Entropy factor
● Smooth function varies on scale of 1-site energies
● If then all is as if fully random
Example: eigenstates are random vectors RMT Take arbitrary vector , then with large probability
In particular, choose , then
Many-Body Localization (MBL)
● Definition: Robustly Non-thermalizing (not just setting hopping = 0)
● Main Example: disordered XXZ:
● Theory (Basko, Aleiner, Altschuler ‘05, Serbyn,Papic,Abanin ‘13, Huse, Oganesyan ‘13, Imbrie ‘14):
sole change
There is quasilocal unitary transformation s.t.
? Numerics (up to 22 sites) (Alet, Laflorencie, Luitz 16)
thermalizing MBL
Local Integrals of Motion (LIOMs)
(Localization length)
● Existence of LIOMs no thermalization, no ETH
● LIOMs like action variables in KAM theory: “ new type of integrability ”
● Non-interacting fermions:
● i.e. LIOMs = number operators of one-particle eigenmodes
quasilocal unitary transformation such that
Stability of MBL wrt Ergodic Grains
Ergodic Grain:ETH
Consider a small ergodic grain (finite bath) in a localized material: What happens?
Assume grain has ETH (Even more: Random Matrix Theory)
Assume localized material has Exponentially local LIOMs
Relevant for realistic materials (large low-disorder Griffiths regions)
Building the Model: Ergodic Grain + MBL
Weak disorder Strong disorder
Model by GOE Matrix (Bath) Model by LIOMs: Spin-coupling terms eliminated
Bath
Unitary
Bath
Bath-LIOM coupling exp. decaying
Local coupling
MBL + Ergodic Grain: Simplest Model
loc length
Random fields
GOE Matrix
BathBath
‘l-bits’ or ‘LIOMs’
Exponential decay of couplings is due to exponential tails of LIOMs
However, GOE-Matrix bath breaks integrability: Model is interacting
MBLMBL Bath-MBL couplingBath-MBL coupling
bath
Strategy to couple 1 spin to bath : Thouless parameter
: spin remains localized (perturbation theory applies)
: spin thermalized: spin ‘joins the bath’
Calculation: Get matrix element of by Random Matrix Theory or ETH
Conclusion: Of course large bath thermalizes spin (if not ridiculously small)
More standard question: thermalization rate:does not scale with dimension (volume)
GOE Matrix
Main Assumption: When a spin joins the bath, we get a new bath that is again GOE:
● Hence dim(B) grows, easier to thermalize next spins
● But coupling to further spins decreases by design
● Competition between these two effects captured by flow of Thouless parameter
Hence all depends on whether or
Stable scenario: Most of chain still localized
Ergodic grain Ergodic grain
Full melted region Full melted region
Avalanche scenario: No MBL but still very slow dynamics
Ergodic grain
Melted region invades whole chain
local thermalization rate
What in other geometries?
Grain
Grain
Grain
Grain
Critical on growth Hilbert space dimension with distance
No stable MBL in d=2
Localized:
Setup for numerics
Ergodic:
More precise RMT yields:
GOE
3 spins 13 spins
The smaller , the better LIOM (or site) i is thermalized
(average over states and disorder°
Numerical study of Var(i) confirms theory in remarkable way
● Spins near bath thermal.
● Spins far from bath go to perfect localization (Var = 1)
● Almost no dependence on size L
● Every spin i becomes perfectly thermal as L grows: Var(i) 0
● Compare each last spin as function of L:more thermal as L grows
Conclusion up to now● MBL in 1d: Strict bound on loc. Length (at least at T=∞)
● No MBL in 1d with long-range interactions
● No MBL in d>1, no matter how strong the disorder
● Of course, still very long thermalization times (Quasi-localization)
Bath
Bath
Bath
Density of ergodic grains:
n= # resonant sites to make bath
Distance between grains:
Thermalization time:
See avalanches in XXZ model directly? (in progress)● Some papers (Goihl, Eisert, Krumnow 2019) and experiments (Bloch lab Munich)
question the avalanche scenario in interacting chains and 2d
● Our finding: Influence of
ergodic grain
does not vanish
as distance
Var(i) with ergodic grain
Var(i) with no ergodic grain
Asymptotic MBL aka. quasi-MBL
● Definition: System is almost brought into LIOM form:
● ‘Small terms’ (in some sense) are often non-perturbative effects, leading to highly suppressed transport and very slow thermalization
● At the heart is always a local absence of resonances (cf KAM theorem)
● Examples:
● Some analogies with glassy dynamics: ‘Jamming of resonances’
* Bose Hubbard model (no disorder) at high density* Classical disordered oscillator chains at large disorder* Classical rotor chains (no disorder) at high energy* ………..
Example: Classical Rotor Chain
● Hamiltonians of angles and angular velocities
● Resonance only if
● Away from resonances, KAM theorem applies
● At small є resonances are rare in Gibbs state at positive temperature
● Even if resonance of 2-3 neighbours, perhaps this remains isolated island?
So maybe: this system is exactly MBL?
No, we do not believe that (because resonant spots should be mobile)
But it surely is Asymptotic MBL!
Asymptotic MBL in rotor chain
Split Ham in local terms and define fluctuations
Theorem: Fluctuations frozen up to very long times
Strongly suggests that also conductivity smaller than
Example: Periodic Driving
local (many-body) Hamiltonians chain of length L
Evolution after
…… should heat up to infinite temp.
Possible obstruction:
some local Ham
Obstruction…but usually also prethermalization
Possible obstruction:
some local Ham
Prethermal state: “Quasi-stationary Noneq state” (Berges, Gasenzer, 2008-…)
is analogue of (a single) LIOM
Initial state Trace state (featureless)Equilibrium state determined by : “Prethermal state”
Simplest example of obstruction: high frequency
Baker-Campbell-Hausdorf?No, converges only for
Still, can construct
Prethermalization up to exponential times!
(Magnus, …..D’Alesio et al,….. Rigourous 2017: Kuwahara et al, Abanin et al )
Kapitza’s Pendulum
Evidence for many-body ergodicity
● For translation-invariant systems: direct from clustering (Keating at al 2013)
● No distinction between integrable and non-integrable systems
Weak ETH ETH Thermalization
● No proof
● Numerical evidence (Rigol et al 2008-…)
● As stated, true for free fermions and certain interacting integrable models
● If one refines the definition, feels equivalent to ETH
Note: Weak ETH Ergodicity
(because non-equilibrium initial state is by definition exceptional entropically)