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

Life beyond dichotomy thermalization/MBL

MBL

Thermalization

Integrable models

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

Thanks for your attention!

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)