Many-Body Localization Wojciech De Roeck, Leuven Francois Thimothée Markus David Luis Huveneers Thiery Müller Luitz Colmenarez
Quantum spin chains I Hilbert space Hamiltonian ● Time evolution ● is conserved: restrict to sector ●
Quantum spin chains I Hilbert space Hamiltonian ● Interaction: Free fermions: Fermion number operator:
Quantum spin chains II Eigenstates ● Equilibrium ensemble ● (microcanonical shell) Stay away from spectral edges: extensive entropy! ● 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: ● ? Numerics (up to 22 sites) thermalizing MBL (Alet, Laflorencie, Luitz 16) sole change Theory ( Basko, Aleiner, Altschuler ‘05 , Serbyn,Papic,Abanin ‘13, Huse, Oganesyan ‘13, Imbrie ‘14 ): ● There is quasilocal unitary transformation s.t.
Local Integrals of Motion (LIOMs) quasilocal unitary transformation such that (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
Life beyond dichotomy thermalization/MBL MBL Thermalization Integrable models
Stability of MBL wrt Ergodic Grains Consider a small ergodic grain (finite bath) in a localized material: What happens? Ergodic Grain: ETH 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 Strong disorder Weak disorder Model by LIOMs: Model by GOE Matrix (Bath) Local coupling Spin-coupling terms eliminated Unitary Bath Bath-LIOM coupling exp. decaying Bath
MBL + Ergodic Grain: Simplest Model Random fields l oc length Bath MBL Bath-MBL coupling Bath MBL Bath-MBL coupling ‘l-bits’ or ‘LIOMs’ GOE Matrix Exponential decay of couplings is due to exponential tails of LIOMs However, GOE-Matrix bath breaks integrability: Model is interacting
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? Critical on growth Hilbert space dimension with distance Grain Grain Grain Grain No stable MBL in d=2
Setup for numerics GOE 13 spins 3 spins (average over states and disorder° Localized : Ergodic : More precise RMT yields: The smaller , the better LIOM (or site) i is thermalized
Numerical study of Var(i) confirms theory in remarkable way ● Spins near bath thermal. ● Every spin i becomes perfectly thermal as L grows: Var(i) 0 ● Spins far from bath go to perfect localization (Var = 1) ● Compare each last spin as function of L: ● Almost no dependence on size 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 no ergodic grain Var(i) with 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) * Bose Hubbard model (no disorder) at high density Examples : ● * Classical disordered oscillator chains at large disorder * Classical rotor chains (no disorder) at high energy * ……….. Some analogies with glassy dynamics: ‘Jamming of resonances’ ●
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 Equilibrium state determined Trace state (featureless) Initial state by : “ Prethermal state ” Prethermal state: “Quasi-stationary Noneq state” (Berges, Gasenzer, 2008-…) is analogue of (a single) LIOM
Simplest example of obstruction: high frequency Baker-Campbell-Hausdorf? No, converges only for Still, can construct Kapitza’s Pendulum Prethermalization up to exponential times! (Magnus, …..D’Alesio et al,….. Rigourous 2017: Kuwahara et al, Abanin et al )
Thanks for your attention!
Evidence for many-body ergodicity Weak ETH ETH Thermalization As stated, true for No proof ● For translation- ● ● free fermions and invariant systems: certain interacting direct from clustering integrable models (Keating at al 2013) Numerical evidence ● (Rigol et al 2008-…) If one refines the ● No distinction ● definition, feels between integrable equivalent to ETH and non-integrable systems Note: Weak ETH Ergodicity (because non-equilibrium initial state is by definition exceptional entropically)
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