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Many-Body Localization in the Age of Classical Computing

Piotr Sierant, Maciej Lewenstein, Antonello Scardicchio, Lev Vidmar, Jakub Zakrzewski

arXiv:2403.07111v3cond-mat.dis-nncond-mat.quant-gascond-mat.stat-mechcond-mat.str-elquant-ph

TL;DR

Whether many-body localization persists as a true phase in the infinite-size, infinite-time limit remains unresolved. This review synthesizes recent classical numerical studies, finding persistent finite-size drifts toward ergodicity while strong-disorder dynamics slow abruptly, leaving the MBL phase status open.

  • Problem

    Whether the MBL regime becomes a phase without asymptotic thermalization as system size and evolution time diverge remains unresolved.

    Source: A. Main open questions · pp. 11–12

  • Method

    The review synthesizes recent numerical investigations of disordered many-body systems, emphasizing calculations on classical computers and exact-diagonalization approaches.

    Source: A. Main open questions · p. 11, A. Exact diagonalization · p. 20

  • Results

    Persistent finite-size drifts toward ergodicity prevent unambiguous extrapolation to the asymptotic limit, although dynamics slow abruptly as disorder increases.

    Source: A. Summary · p. 74

  • Takeaways & Limitations

    The status of the MBL phase and the ETH-MBL transition remains unresolved, with microscopic details potentially determining the balance between ergodicity and localization.

    Source: A. Summary · pp. 74–75

    Available numerical results do not determine whether or at what disorder strength a transition to the MBL phase occurs.

    Source: A. Summary · p. 74

Abstract

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Statistical mechanics provides a framework for describing the physics of large, complex many-body systems using only a few macroscopic parameters to determine the state of the system. For isolated quantum many-body systems, such a description is achieved via the eigenstate thermalization hypothesis (ETH), which links thermalization, ergodicity and quantum chaotic behavior. However, tendency towards thermalization is not observed at finite system sizes and evolution times in a robust many-body localization (MBL) regime found numerically and experimentally in the dynamics of interacting many-body systems at strong disorder. Although the phenomenology of the MBL regime is well-established, the central question remains unanswered: under what conditions does the MBL regime give rise to an MBL phase in which the thermalization does not occur even in the asymptotic limit of infinite system size and evolution time? This review focuses on recent numerical investigations aiming to clarify the status of the MBL phase, and it establishes the critical open questions about the dynamics of disordered many-body systems. Persistent finite size drifts towards ergodicity consistently emerge in spectral properties of disordered many-body systems, excluding naive single-parameter scaling hypothesis and preventing comprehension of the status of the MBL phase. The drifts are related to tendencies towards thermalization and non-vanishing transport observed in the dynamics of many-body systems, even at strong disorder. These phenomena impede understanding of microscopic processes at the ETH-MBL crossover. Nevertheless, the abrupt slowdown of dynamics with increasing disorder strength suggests the proximity of the MBL phase. This review concludes that the questions about thermalization and its failure in disordered many-body systems remain a captivating area open for further explorations.

I. INTRODUCTION … C. MBL phase: a definition

This section introduces MBL as a disorder-induced failure of thermalization in interacting quantum systems and frames its phase definition through asymptotic non-ergodic behavior. It discusses the challenges of distinguishing ergodic and MBL systems, including size-dependent evolution times and debated criteria involving LIOMs and transport.

II. MBL: HISTORICAL PERSPECTIVE IN SHORT

  • MBL research combines Anderson localization, quantum chaos, ETH, random-matrix methods, and entanglement measures to study disordered interacting systems at high temperature.The field developed from initially separate studies of localization, spectral statistics, thermalization, and quantum information, with numerical simulations remaining central despite experimental progress.Source: §II · p. 4

A. Quantum thermalization

  • ETH makes long-time few-body observables depend on conserved quantities through microcanonical averages, whereas its validity is established as sufficient for thermalization but not proved for generic local Hamiltonians.The equilibrium value is controlled by diagonal eigenbasis matrix elements, while off-diagonal elements determine the approach timescale, which can be exponentially long in system size.Source: A. Quantum thermalization · pp. 5–6, B. Ergodicity breaking: single… · p. 6

B. Ergodicity breaking: single particle systems and beyond

  • Anderson localization provides a single-particle counterexample in which strong disorder confines particles, eliminates conduction, and invalidates recovery of local observables by microcanonical averages.Whether interactions destroy this localization by generating conductivity remained the key question; perturbative work proposed that localization persists below a critical temperature at weak interactions.Source: B. Ergodicity breaking: single… · p. 6

C. MBL phase: a definition

  • The MBL phase is defined as robust disorder-induced non-ergodicity for physically relevant initial states and real-space local observables, excluding fine-tuned integrable models and many-body scars.Ergodicity instead requires thermalization for all local observables; when both phases exist, an MBL transition separates them.Source: C. MBL phase: a definition · p. 7
  • MBL requires persistent local-observable deviations from thermal predictions as t →∞ and L →∞, with eigenstate fluctuations expected not to decay with system size for most eigenstates.Exact diagonalization takes the infinite-time limit first, yielding A∞(L), which must then be extrapolated to infinite system size; the stability of this phase remains unproved.Source: C. MBL phase: a definition · p. 7
  • The thermodynamic limit is normally taken at fixed Hamiltonian parameters, although scaling microscopic parameters with system size can produce other dynamical phenomena such as those from long-range couplings.Studies of thermalization predominantly use product or low-entangled initial states, and the definition can extend to different few-body observables.Source: C. MBL phase: a definition · pp. 7–8
  • Time scales needed to distinguish ergodicity from MBL diverge with system size and can grow as t ∝ L^z for subdiffusive dynamics, making finite-time assessments unreliable.For diffusive dynamics, local observables must be probed at times t = aL^2; probing at t ∝ L can falsely suggest a lack of thermalization.Source: C. MBL phase: a definition · p. 8
  • LIOMs underpin much of the MBL phenomenology, including vanishing conductivity, absent diffusion, and slow entanglement growth, while recent work proves these consequences only under limited conditions.Roeck et al. (2024) claim LIOMs in a fraction of samples exponentially small in system size and prove absent diffusion in a strongly disordered quantum Ising chain, but not an MBL phase.Source: C. MBL phase: a definition · p. 8
  • The review’s MBL definition is not formulated through transport or vanishing DC conductivity and applies to systems without a natural non-interacting limit.Transport properties can instead be derived from local conserved operators, while the definition explicitly allows a broader class of many-body systems than perturbative connections to Anderson localization.Source: C. MBL phase: a definition · p. 8
  • A precise MBL definition remains contested because LIOMs may be unnecessarily restrictive, whereas the stated criterion includes Fock-space localization but also allows eigenstates delocalized on the Fock-space graph.The review notes that its criterion may hold when many-body eigenstates are not Fock-space localized.Source: C. MBL phase: a definition · p. 9

D. Modeling and phenomenology of MBL … B. Difficulties in numerical approaches to MBL

The disordered XXZ chain exhibits strong-disorder phenomenology associated with MBL, including memory retention, slow dynamics, Poisson spectra, and area-law eigenstates, but finite-size and finite-time results do not establish an asymptotic MBL phase. Numerical drifts toward thermalization, uncertain extrapolations, and unresolved transition mechanisms leave the existence, definition, and scope of MBL open.

D. Modeling and phenomenology of MBL

  • The random-field XXZ chain shows non-ergodic phenomenology including initial-state memory, logarithmic entanglement growth, dephasing, logarithmic light cones, Poisson spectra, multifractal eigenstates, and area-law entanglement.These observations motivated interpreting strong-disorder behavior as a stable MBL phase.Source: D. Modeling and phenomenology of MBL · p. 9
  • LIOMs explain memory retention, absent transport, Poisson level statistics, area-law eigenstates, and slow information spreading, but their construction remains approximate or restricted to small systems.Perturbative and numerical constructions do not settle the limits t →∞ and L →∞; multiscale constructions require convergence and locality that remain contested.Source: D. Modeling and phenomenology of MBL · p. 10
  • The avalanche mechanism models an ergodic seed coupled to localized qubits, but its relevance to disordered chains and higher-dimensional systems remains unclear.Quantum sun models nevertheless provide a reference transition whose numerical critical point agrees closely with the analytical hybridization prediction.Source: D. Modeling and phenomenology of MBL · p. 10

E. Challenges to MBL

  • Finite-time and finite-size observations of the XXZ chain support an MBL regime, but slow creep toward thermalization and competing interpretations leave stable MBL unresolved.The MBL regime denotes finite systems with non-ergodicity when one or both asymptotic limits cannot be properly taken.Source: E. Challenges to MBL · pp. 10–11

IV. STATUS OF MBL

  • Classical simulations typically reach only 6–20 or 22 qubits or roughly a hundred cycles, so extrapolation to t →∞ and L →∞ is nonunique.Proposed transition theories have not yet been demonstrated relevant to the numerically and experimentally studied prototype models.Source: §IV · p. 11

A. Main open questions

  • Open questions concern defining MBL through the joint long-time and thermodynamic limits, relating MBL to LIOMs and Fock-space localization, and identifying decisive signatures.They also include whether ETH can coexist with subdiffusion, whether transport can persist in MBL, and how avalanches and resonances drive the transition.Source: A. Main open questions · pp. 11–12

B. Difficulties in numerical approaches to MBL

  • Spectral indicators should cross at an L-independent disorder in an ideal transition, but one-dimensional local systems show crossing drifts toward larger W that may continue indefinitely.Thus finite-size crossovers can represent an MBL transition or an ultimately ergodic phase, depending on the thermodynamic extrapolation.Source: B. Difficulties in numerical… · p. 13, Figure 2 · p. 13
  • Time-domain observables decay more slowly at stronger disorder but also more persistently with increasing size and time, preventing distinction between MBL and eventual thermalization.Across observables, persistent thermalization trends coexist with extremely slow dynamics, making the existence and location of MBL difficult to determine.Source: Figure 3 · p. 13, B. Difficulties in numerical… · p. 14

C. Remarks on finite-size scaling … 3. Unconventional scaling limits

Finite-size scaling at the ETH–MBL crossover is difficult because persistent crossing-point drifts undermine single-parameter interpretations. Sub-leading corrections and unconventional scaling limits offer alternatives, but available system sizes often cannot distinguish competing scenarios or establish an asymptotic MBL phase.

C. Remarks on finite-size scaling

  • Finite-size scaling is central to determining whether MBL exists as a phase and to extracting the critical properties of the ETH–MBL transition.The analysis uses the Anderson transition as a reference while emphasizing that numerical crossover results require careful scrutiny.Source: C. Remarks on finite-size scaling · p. 14

1. Single parameter scaling

  • Single-parameter scaling fails to reproduce the significant drift of the crossing disorder observed at the ETH–MBL crossover.With a continuous, monotonic scaling function, crossings for different sizes are forced to occur at one critical disorder, contrary to the observed drift.Source: §1 · p. 15
  • Ignoring crossing-point drift can produce appealing data collapses but yields ν close to 1, contradicting the 1D Harris–Chayes bound ν ≥2.Reported fits include Kjäll et al. (2014), Luitz et al. (2015), and Pal and Huse (2010), while the bound is attributed to Chayes et al. (1986) and Harris (1974).Source: §1 · p. 15
  • Power-law and BKT scaling are difficult to distinguish because BKT finite-size convergence is much slower, even across model sizes reaching L ≈10^5.A toy-model analysis was first interpreted as power-law scaling with ν ≈3, then refined to BKT scaling; available many-body sizes span less than two orders of magnitude.Source: §1 · p. 16

2. Sub-leading corrections to the scaling

  • Including an irrelevant scaling variable introduces sub-leading corrections that allow the crossing point to drift with system size.The correction depends on the irrelevant exponent y, the relevant exponent ν, and the scaling functions, making this ansatz better suited to ETH–MBL crossover data.Source: §2 · p. 16
  • Although corrected scaling predicted ν ≈2 for a 1D interacting disordered Floquet system, its many free parameters and limited sizes severely restrict predictive power.Even random-regular-graph data with known critical disorder cannot clearly determine ν from currently available numerical results.Source: §2 · p. 17

3. Unconventional scaling limits

  • An XXZ-chain study found BKT collapses with a transition point increasing linearly, f W_C(L) = a_0 + a_1L, which would imply no conventional finite-disorder MBL transition.This result may instead reflect accessible-size behavior that eventually crosses over to a finite critical disorder, so its thermodynamic implication remains unresolved.Source: §3 · p. 17
  • Two-dimensional Anderson localization shows that apparently successful collapses can mask unconventional limits, cautioning against naive power-law scaling for MBL.Related unconventional limits also produce ergodicity-breaking transitions in power-law-interaction chains and avalanche models when the thermodynamic limit is taken with size-dependent scaling variables.Source: §3 · pp. 17–18

D. Conclusion … C. Tensor network and other approaches

The review finds that finite-size drifts toward ETH keep the asymptotic status of the MBL phase unresolved, while numerical methods trade exactness and accessible scales against exponential Hilbert-space growth or entanglement limitations.

D. Conclusion

  • Persistent drifts toward ETH with increasing time and system size prevent distinguishing slow dynamics from complete arrest, leaving the existence of an asymptotic MBL phase unresolved.Single-parameter scaling is insufficient at the ETH-MBL crossover, requiring sub-leading corrections through irrelevant variables or unconventional scaling functions.Source: D. Conclusion · p. 18

V. NUMERICAL METHODS FOR MBL

  • Numerical MBL studies combine exact diagonalization, sparse-matrix time evolution, and tensor-network approximations to probe eigenstates, thermalization, and dynamics across accessible sizes and times.Exact diagonalization targets highly excited eigenstates, while sparse Hamiltonian methods enable numerically exact time evolution beyond full diagonalization limits.Source: §V · pp. 18–19, Table I · p. 19

A. Exact diagonalization

  • Full exact diagonalization is limited by exponential Hilbert-space growth, typically reaching N ∼10^5 and restricting disordered XXZ calculations to L ≤18 after disorder averaging.Sparse matrices enable Lanczos-based methods, but shift-and-invert calculations incur substantial memory overhead because LU decomposition destroys sparsity.Source: A. Exact diagonalization · pp. 19–20
  • POLFED preserves Hamiltonian sparsity and needs ≈100 GB of RAM for 2000 middle-spectrum eigenvectors at L = 24, versus ≈10^4 GB for shift-and-invert, with similar CPU time.Polynomial spectral transformations also extend Floquet eigenvector calculations beyond full diagonalization, reaching L = 20 or more and N > 10^6.Source: A. Exact diagonalization · pp. 20–21

B. Time evolution methods

  • Chebyshev and Krylov propagation exploit Hamiltonian sparsity for numerically exact dynamics, reaching N ≈10^9 and t ≈10^3 on single nodes, or N ≈10^10 with internode communication.Chebyshev propagation is a couple of times faster than Suzuki-Trotter schemes, while Krylov evolution computes exponentials in a small dynamically expanded subspace.Source: B. Time evolution methods · p. 21

C. Tensor network and other approaches

  • Tensor-network methods represent some one-dimensional states with polynomially many parameters and simulate hundreds of sites, but their efficiency fails when entanglement grows toward a volume law.For realistic bond dimensions, chaotic dynamics remain accurate for about 10 tunneling times, whereas strong disorder can extend simulations to a few thousand tunneling times through slowed, logarithmic entanglement growth.Source: C. Tensor network and other approaches · p. 22
  • Light-cone renormalization and numerical linked-cluster expansions exploit locality or discrete disorder to study dynamics directly in the thermodynamic limit L →∞.These approaches use Lieb-Robinson light cones and translationally invariant enlarged local spaces to investigate ergodic and MBL quenches.Source: C. Tensor network and other approaches · p. 22
  • Spectral-transformation and density-matrix-renormalization approaches target putative area-law eigenstates, but their bias toward low entanglement may miss precursors of thermalization.Low-depth circuit methods extend studies to experimentally relevant two- and three-dimensional systems, although their reliable size and time ranges remain uncertain.Source: C. Tensor network and other approaches · pp. 22–23

D. Outlook … 2. Thouless time

This section examines numerical and analytical approaches to the ETH–MBL crossover and the unresolved existence of an asymptotic MBL phase. It focuses on finite-size spectral and dynamical scaling, the disorder-dependent Thouless time, and contrasting scenarios for the thermodynamic limit.

D. Outlook

  • Exact diagonalization accesses all eigenstates but is exponentially limited in L, whereas shift-and-invert, POLFED, Chebyshev/Krylov, and tensor-network methods extend spectral or dynamical calculations.ED and Chebyshev/Krylov schemes provide numerically exact results useful near the ETH–MBL crossover, while tensor networks reach larger sizes if convergence is monitored.Source: D. Outlook · p. 23
  • Other approaches—including LIOM construction, MCLM, open-system tensor networks, and machine learning—broaden available probes but machine-learning transition estimates remain sensitive to architecture and training parameters.Machine learning is nevertheless useful for analyzing disordered-system dynamics, classifying non-ergodic eigenstates, and identifying quantum-many-body phases.Source: D. Outlook · p. 23

1. Gap ratio

  • Finite-size gap-ratio drifts are compatible with either a finite-disorder MBL transition or ergodicity at every disorder strength as L →∞.For the J1-J2 model, weak disorder trends toward rGOE, strong disorder trends toward rP S, and present numerics cannot distinguish the two scenarios; their scalings become incompatible near L0 ≈50 in the XXZ chain.Source: §1 · pp. 24, 26

2. Thouless time

  • The Thouless time marks when the spectral form factor follows GOE ramp-and-plateau behavior and links long-range spectral correlations to relaxation below the Heisenberg time.It is defined as tT h = τT htH using a fixed tolerance ϵ = 0.05; in the J1-J2 model, τTh decreases with L in the ergodic regime while tTh grows approximately exponentially with disorder.Source: Figure 5 · p. 26, §2 · p. 27
  • The two-parameter Thouless-time hypothesis implies GOE behavior below W T Th and therefore no MBL phase if W T Th grows with L.When tTh approaches or exceeds tH, short-range statistics deviate from GOE; the resulting W T Th is closely related to the gap-ratio threshold W T (L).Source: §2 · p. 28
  • Increasing disorder makes the Thouless time grow approximately exponentially, preventing finite systems from clearly distinguishing slow thermalization from asymptotically arrested dynamics.The expected scaling may fail when tTh approaches tH, and available system sizes do not yield unambiguous conclusions at large W.Source: §2 · p. 28
  • The diffusive scaling tTh ∝ L^2 may break down at large L and W, while subdiffusive forms tTh ∝ L^z with z > 2 leave the existence of the MBL phase unresolved.Thus, the scaling for disordered spin chains does not by itself rule out an MBL phase in the asymptotic limit (Šuntajs et al., 2020; Sierant et al., 2020a,b).Source: §2 · p. 28
  • In generic ergodic Floquet models without local conservation laws, the Thouless time scales logarithmically with system size, and randomized-measurement protocols have been proposed and implemented for small systems.The logarithmic dependence is supported by transfer-matrix calculations but is difficult to observe in exact diagonalization (Chan et al., 2018b; Kos et al., 2018; Garratt and Chalker, 2021a,b; Sonner et al., 2021; Joshi et al., 2022; Dong et al., 2024).Source: §2 · p. 28

3. Other measures of spectral statistics … 1. Wave function properties

Spectral statistics and eigenstate structure provide complementary indicators of ergodicity breaking, but finite-size drifts and methodological sensitivities prevent unambiguous identification of an asymptotic MBL phase. Wave functions are multifractal in the MBL regime, while long-range spectral probes and entanglement show strong disorder-dependent slowdowns and crossover shifts.

3. Other measures of spectral statistics

  • The level spacing distribution changes with system size at fixed disorder, while its extraction depends on non-unique unfolding; therefore, average gap ratios are more commonly used.Unfolding can produce misleading long-range spectral-statistics conclusions, motivating the simpler average gap ratio.Source: §3 · pp. 28–29
  • Higher-order spacing ratios probe larger energy scales, can assess RMT properties without resolving Hamiltonian symmetries, and exhibit linear system-size drifts at the ETH-MBL crossover.The number variance and related long-range statistics provide alternative definitions of the Thouless time.Source: §3 · p. 29

4. Overview

  • The average gap ratio probes the Heisenberg-time scale and is suited to distinguishing ergodic and MBL behavior, but crossover shifts with system size make present numerical conclusions ambiguous.Level spacing distributions show similar system-size flows, so alternative spectral measures do not remove the central finite-size limitation.Source: §4 · p. 29
  • The spectral form factor’s Thouless time grows approximately exponentially with disorder, yet accessible sizes remain consistent with either slow thermalization or complete dynamical freezing as t, L →∞.Its ramp-plateau structure defines tTh, but scaling with size and disorder yields ambiguous ETH-MBL crossover conclusions.Source: §4 · p. 29

B. Properties of eigenstates

  • Exact diagonalization supplies eigenstates whose basis-dependent structure and quantum-information measures offer additional indicators of ergodicity breaking.The computational basis is natural for disordered spin chains, and eigenstates coincide with basis states as W →∞ at fixed L.Source: B. Properties of eigenstates · p. 29, §1 · p. 29

1. Wave function properties

  • In the J1-J2 model, ETH eigenstates have Dq = 1, whereas MBL eigenstates are multifractal with 0 < Dq < 1 and approximately Dq ∝1/W at W ≫1.Thus MBL eigenstates are not localized in the computational basis; Dq = 0 corresponds to the stronger condition of Fock-space localization.Source: §1 · p. 30
  • The fractal-dimension crossing drifts toward larger W with increasing L, preventing unambiguous conclusions about the ETH-MBL crossover in the infinite-size limit.Dq is non-monotonic in W, its Dq > 1 region shrinks with L, and the crossing drift is weaker than for the average gap ratio.Source: Figure 6 · p. 30, §1 · p. 31
  • Participation entropies require disorder averaging near the crossover because they may not be self-averaging, while multifractal wave functions can support Hilbert-space decimation schemes.Results for Dq remain qualitatively similar in localized Anderson-orbital bases, although dynamics need not depend strongly on eigenstate multifractality.Source: §1 · p. 31

2. Quantum information measures … 1. Spectral function

Quantum-information measures distinguish ETH and MBL eigenstates, but finite-size drifts prevent controlled asymptotic conclusions. Spectral functions connect matrix elements to dynamics, revealing disorder-driven slowdown, loss of RMT plateaus, and persistent ambiguity about the thermodynamic limit.

2. Quantum information measures

  • ETH eigenstates exhibit volume-law entanglement and distance-independent QMI, whereas MBL eigenstates exhibit area-law entanglement and QMI that decays exponentially with subsystem separation.These measures operationalize the contrast between nearly Haar-random ergodic eigenstates and localized eigenstates.Source: §2 · p. 32
  • The rescaled entanglement entropy s approaches unity in ETH and zero in MBL, yet its crossing point drifts strongly with system size, ruling out naive single-parameter scaling.The drift prevents unambiguous conclusions about the ETH-MBL crossover as L →∞.Source: §2 · p. 32
  • Entanglement-entropy fluctuations grow superlinearly across disorder samples in the crossover, but this pre-asymptotic trend cannot persist indefinitely and contributes to scaling ambiguities.The reported behavior is σS ∝L^α with α > 1, while the maximal standard deviation is proportional to L.Source: §2 · pp. 32–33

3. Perspective

  • Eigenstate indicators such as participation entropies and entanglement entropy probe different wave-function properties, and either volume-law entanglement or Page-value saturation can coexist with multifractality.Their similar system-size drifts prevent unambiguous conclusions about disordered spin chains in the joint L,t →∞ limit.Source: §3 · p. 33

C. Matrix elements of local observables

  • Matrix elements of local observables link spectral structure to measurable dynamics, with the spectral function organizing diagonal and off-diagonal ETH diagnostics.Its low-frequency delta-weight reflects diagonal matrix elements, while its conductivity relation applies when the operator is a conserved-density operator.Source: C. Matrix elements of local observables · p. 33, §1 · pp. 33–34

1. Spectral function

  • Increasing disorder narrows the spectral-function plateau, suppresses ωTh/ωH, and increases the high-frequency decay exponent, signaling progressively slower dynamics before the MBL regime.Near the ETH-MBL crossover, α approaches unity, corresponding to extremely slow power-law or logarithmic autocorrelation decay.Source: §1 · p. 34
  • In the MBL regime, the averaged spectral function loses its plateau, while stiffness C0 becomes approximately system-size independent instead of decaying exponentially as in ETH.These contrasting behaviors distinguish ergodic and localized dynamics, but their finite-size trends do not determine the L →∞ limit.Source: §1 · pp. 34–35, Figure 8 · p. 34
  • A broad distribution of relaxation times, including LIOMs with tR exceeding the Heisenberg time, may control the ETH-MBL crossover through competition with the exponentially growing tH.This mechanism is proposed from the non-interacting limit, but understanding the disorder evolution of the spectral function remains unresolved.Source: §1 · p. 35

2. Sensitivity to local perturbations … 1. Density correlation functions

This section examines spectral, eigenstate, and density-correlation responses to local perturbations as probes of ergodic and MBL behavior. It focuses on finite-size effects, slowing dynamics, persistent local memory, and the unresolved question of asymptotic thermalization.

2. Sensitivity to local perturbations

  • Level dynamics distinguish regimes: ergodic systems show avoided crossings from global correlations, whereas MBL systems show regular motion and rare crossings associated with localized constants of motion.Rare avoided crossings can be interpreted as resonances between LIOM configurations occurring over increasing length scales as disorder decreases toward the ETH-MBL crossover.Source: §2 · p. 36
  • Fidelity susceptibility reproduces ETH-MBL crossover drifts rather than resolving the transition: χtyp ∝2^L in ETH, χtyp ∝4^L in the maximally chaotic regime, and its crossing drifts with L.The rescaled χtyp/2^L crosses near W*≈4.5 and the maximum occurs at Wm∝L for the studied sizes; these trends are consistent with other drifting indicators.Source: §2 · p. 36
  • Fidelity-susceptibility tails require matrix-element divergences and correlations with energy denominators, invalidating an assumed independence used to interpret P(x)∝x^-4 as emergent level attraction.These correlations and small-frequency spectral-function divergences should be incorporated into phenomenological descriptions of the MBL phase.Source: §2 · p. 37

3. Other approaches and outlook

  • Matrix elements of local observables retain ETH-MBL crossover drifts, while their spectral function links low-frequency plateaus to Thouless energy; the relations among Thouless-time definitions remain unresolved.The Binder cumulant shows no clear transition signature, although extrapolation places a Wc=10 transition beyond L0≈57; related observables likewise do not settle the thermodynamic limit.Source: §3 · pp. 37–38

1. Density correlation functions

  • Density autocorrelations decay to zero in the ergodic regime but retain nonzero long-time memory in MBL, while increasing disorder abruptly slows dynamics and leaves asymptotic thermalization unresolved.For W=4 and Δ=0, Anderson localization produces saturation after initial oscillations; for W=5, interacting dynamics decays persistently but very slowly, with the L,t→∞ behavior unknown.Source: §1 · p. 39, Figure 10 · p. 39
  • At W = 4, interactions produce a slowly decaying C(t) that persists until approximately the Heisenberg time t_H ∝ e^cL, so finite-time behavior may strongly affect asymptotic conclusions.The persistent decline is attributed to interactions rather than a time-independent plateau.Source: §1 · p. 40
  • At W = 3, C(t) remains above 0.4 for 12 ≤ L ≤ 20 but decreases approximately proportionally to L, extrapolating to thermalization as L, t →∞.These finite systems are nonergodic and lie in the crossover regime W_T(L) < W < W*(L), despite the eventual decay inferred from extrapolation.Source: §1 · p. 40
  • Spatial disorder fluctuations generate Griffiths regions that bottleneck thermalization and produce slow power-law local-correlator decay, while resonant regions may permit subdiffusive behavior at strong disorder.The observed power laws were linked to heterogeneities in disordered systems and have also appeared in survival-probability dynamics.Source: §1 · p. 40
  • For W ≳ 3, extrapolation becomes unclear: at W = 5 and L ≤ 20, C(t) decays more slowly than logarithmically, while larger systems restore prominent power-law decays.Chebyshev and tensor-network studies extended accessible sizes to L = 24 and L = 100, respectively, and interpreted the behavior as an MBL transition above early exact-diagonalization estimates.Source: §1 · p. 40
  • Any fixed-exponent decay C(t) ∝ t^−β persisting to t_H gives C(t_H) ∝ e^−βcL, implying thermalization regardless of how small β is.Consequently, asymptotic MBL dynamics must decay more slowly than any power law.Source: §1 · p. 40

2. Assessing the breakdown of ergodicity … 1. Logarithmic growth of entanglement entropy

Disordered XXZ chains exhibit exponentially slowed, apparently nonergodic dynamics, but finite-size and finite-time trends prevent distinguishing an asymptotic MBL phase from eventual thermalization. Entanglement growth and spatial correlations similarly show slow behavior that remains difficult to classify asymptotically.

2. Assessing the breakdown of ergodicity

  • In the disordered XXZ spin-1/2 chain, the decay exponent β decreases exponentially with disorder strength, β ∝ e−γW, across W ∈ [1, 10]; at W = 8, the density autocorrelation C(t) fits β ≈ 10−3.Source: §2 · p. 41, Figure 11 · p. 41
  • The observed slow autocorrelation decay is compatible both with an MBL phase, where C(t) saturates nonzero, and with eventual thermalization if the decay persists.Present numerical results do not distinguish scenarios in which crossover disorders converge to a finite critical value or diverge with system size.Source: §2 · p. 41
  • At β ≈10−3, C(t) would require approximately 10^1000 tunneling times to reach 10% of its initial value, making the large-disorder dynamics effectively arrested in practice.This extreme slowdown is robust numerically even though it does not resolve whether thermalization ultimately continues or stops.Source: §2 · p. 42

3. Spatial decay of correlations

  • Spatial correlations remain nearly exponential in distance, while the dynamical exponent z decreases toward diffusion with increasing size, preventing a firm conclusion that subdiffusion is asymptotic.These size-dependent trends occur even in the MBL regime and agree with thermalizing tendencies found from autocorrelation dynamics.Source: §3 · p. 42
  • Local-observable dynamics distinguish accessible ergodic and nearly arrested MBL regimes, but their exponential slowdown with disorder leaves the asymptotic fate ambiguous.Time evolution is numerically feasible beyond exact-diagonalization sizes, yet only over limited time intervals.Source: §3 · p. 42

B. Entanglement growth

  • Entanglement growth provides a complementary probe of ergodicity, with experiments observing slow spreading in disordered many-body systems and results showing trends toward thermalization.These trends prevent pinpointing the status of the MBL phase, paralleling conclusions from local-observable autocorrelations.Source: B. Entanglement growth · p. 42

1. Logarithmic growth of entanglement entropy

  • In the MBL regime, S(t) is expected to grow logarithmically, but in the crossover it grows sub-ballistically as S(t) ∝tγ, with γ showing slight systematic increases at larger scales.This increase opens the possibility of gradual slow thermalization, while limited time intervals make logarithmic and algebraic growth difficult to distinguish.Source: §1 · p. 43
  • At W = 10, S(t) is better fit by S(t) ∝t^0.08 than by a logarithm over broad times, although longer times could favor logarithmic growth.The observed curvature also appears in an LIOM model possessing MBL features, so the fit does not conclusively establish thermalization or exclude an MBL phase.Source: Figure 12 · p. 43, §1 · pp. 43–44

2. Symmetry resolved entanglement: number entropy · C. Extracting universal features of the slow dynamics · 1. Proximity to Anderson Insulator

The reviewed results reveal ultraslow dynamics across disordered spin chains, while number-entropy growth and Anderson-LIOM relaxation leave the asymptotic ETH–MBL distinction unresolved. Symmetry-resolved entanglement and proximity-to-Anderson analyses identify mechanisms and scaling behaviors that remain consistent with competing long-time scenarios.

2. Symmetry resolved entanglement: number entropy

  • The number entropy S_n tracks fluctuations between symmetry sectors, while configurational entropy S_c captures superpositions within fixed-spin sectors.The U(1) symmetry block-diagonalizes the reduced density matrix, separating these two contributions to entanglement.Source: §2 · p. 44
  • Current numerical evidence supports both persistent ultraslow transport and transient single-particle fluctuations, requiring larger systems and longer times to distinguish them.The competing interpretations respectively imply absence of an MBL phase or compatibility with one.Source: §2 · p. 45
  • S_n(t) ∝ ln ln(t) persists deep in the disordered XXZ chain and in an MBL LIOM circuit, but whether it saturates asymptotically remains unresolved.In the LIOM circuit, this growth persists to L = 28 and times exponentially large in L, while available results still allow saturation at larger scales.Source: §2 · pp. 44–45

C. Extracting universal features of the slow dynamics

  • Slow glassy dynamics, with persistent local-correlator decay and gradual entanglement growth, reaches the largest system sizes and times currently accessible numerically.This motivates extracting universal dynamical features rather than restricting analysis to whether an MBL phase exists strictly as L, t →∞.Source: C. Extracting universal features of… · p. 45

1. Proximity to Anderson Insulator

  • At W = 1.5 and L = 16, interacting and non-interacting spectral functions agree at high frequencies but differ at low frequencies, separating short- and long-time dynamics.The Anderson-localized limit motivates a phenomenological description of the interacting low-frequency spectrum.Source: §1 · p. 45
  • w(τ) ∝ τ^-μ broadens strongly with disorder, producing plateau loss near W ≈ 2 and slow persistent local-correlator decays at larger W.The relaxation-time distribution describes finite Anderson-LIOM lifetimes and the resulting low-frequency spectral behavior.Source: §1 · p. 46
  • The true interaction perturbation decays as 1/W while ||H_0|| ∝ W, making the chain a weakly perturbed Anderson insulator whose dynamics slowly approaches thermal equilibrium.In the MBL regime, τ_min remains below t_H while τ_max exceeds it, leaving some Anderson LIOMs stable and others delocalized.Source: §1 · p. 46
  • Rescaling the perturbation to the non-interacting norm or adding strong pair hopping restores ergodicity at every disorder strength, highlighting perturbation structure as decisive.The results motivate further study of how interactions gradually destabilize Anderson LIOMs and generate slow dynamics.Source: §1 · p. 47

2. Stretched exponential decays … VIII. RELATION OF THERMALIZATION AND TRANSPORT

The review surveys slow dynamics through stretched-exponential relaxation, an entanglement-based internal clock, operator growth, survival probabilities, and transport-oriented probes. Across these diagnostics, finite-size and finite-time results reveal strong slowdowns and nonergodicity-like behavior but no convincing asymptotic evidence establishing an MBL phase.

2. Stretched exponential decays

  • For W ≈ 2.5 near the finite-size ergodic edge, local autocorrelators show stretched-exponential decay with κ < 1 and a characteristic time τ increasing with W.Many-body resonances between locally differing magnetization patterns may explain this regime, while hydrodynamic power-law decay is expected at longer times.Source: §2 · p. 47
  • At W = 3 and L = 20, the stretched-exponential prediction already breaks down, while W = 8 remains nonergodic for practical times despite potentially enormous thermalization times.Jacobi rotations associate the slow dynamics with a broad power-law distribution of resonance frequencies and a hierarchy of successive resonances.Source: §2 · p. 47

3. Internal clock of many-body dynamics

  • For W < 8, plotting disorder-averaged autocorrelation and fluctuation data against entanglement entropy collapses them onto a system-size-dependent master curve, with reduced sample-to-sample fluctuations.The absence of a separate nondecaying branch at t < 10^3 provides no visible MBL signature, motivating larger-time and larger-system studies.Source: §3 · p. 48

4. Operator growth

  • Operator spreading changes from ballistic to algebraic and then logarithmic with increasing disorder, while MBL-like Krylov localization requires even-odd and random Lanczos-coefficient structure.In interacting chains, nested-commutator growth continues accelerating through W = 25 at the probed orders, unlike the at-most-exponential localized bound.Source: §4 · pp. 48–49
  • At order n = 16, corresponding to operators spanning L = 33 sites, nested-commutator growth shows no convincing MBL signature and accelerates beyond the noninteracting case for n ≳ 8.These observations leave quantitative connections between Euclidean-time operator spreading and persistent autocorrelation decay unresolved.Source: §4 · p. 49

5. Dynamics of survival probabilities

  • Survival probabilities decay exponentially small in L in ergodic systems but saturate higher in the MBL regime, while their correlation holes diagnose chaotic dynamics and define an alternative Thouless time.Despite these diagnostics and Fock-space interpretations of slow transport, no sharp prediction for the onset or absence of MBL has been established.Source: §5 · pp. 49–50

VIII. RELATION OF THERMALIZATION AND TRANSPORT

  • Transport in disordered spin chains is investigated through spin-current spectral functions and boundary reservoirs, providing complementary probes of slow dynamics and MBL.The section focuses on selected coherent-transport aspects relevant to relating thermalization, transport, and localization.Source: §VIII · p. 50

A. Transport in the ergodic regime … 3. Non-equilibrium steady states in open systems

Ergodic disordered systems can exhibit ballistic, superdiffusive, subdiffusive, or diffusive transport, while finite-size and finite-time drifts make the transport class near the putative MBL transition difficult to determine. Complementary conductivity, unitary-dynamics, and open-system studies establish diffusion deep in the ergodic regime but leave stronger-disorder transport unresolved.

A. Transport in the ergodic regime

  • Chaotic ergodic systems with random-matrix spectral statistics can support transport other than diffusion, including ballistic and superdiffusive behavior.Examples include impurity systems with ballistic transport and the PXP model with dynamical exponent z ≈3/2.Source: A. Transport in the ergodic regime · p. 51

1. Spin conductivity

  • The diffusion constant in ergodic disordered spin chains decreases exponentially with disorder, D0 ∝e−aW, and RMT deviations emerge when D0 approaches the many-body level spacing.This comparison occurs at disorder strength W ∝L because both the diffusion constant and average level spacing decrease exponentially with their respective variables.Source: §1 · p. 51

2. Unitary time evolution

  • In the disordered XXZ chain at W = 1.5, the dynamical exponent decreases from z ≈4.2 at L = 16 to z ≈2.95 at L = 32, preventing exclusion of diffusion.At W = 3, z likewise drifts from approximately 6.4 at L = 16 to 5.2 at L = 28, while the dynamics slow substantially.Source: §2 · pp. 51–52
  • Three-dimensional Anderson dynamics at W = 15 appear subdiffusive with α ≈0.75 for L = 80 but flow toward diffusion, α →1 and z →2, by L = 120.The result demonstrates that substantial time and length scales can be required to identify transport correctly.Source: §2 · p. 52, Figure 13 · p. 52

3. Non-equilibrium steady states in open systems

  • Across transport probes, stronger disorder slows dynamics and prevents clear classification because finite-size drifts and level-spacing effects obscure whether transport is diffusive, subdiffusive, or localized.Thus, the relation between transport and thermalization, and the fate of the ergodic regime, remains open.Source: §2 · p. 52, §3 · p. 54
  • Tensor-network simulations of nonequilibrium steady states find clear diffusive transport, j ∝L−1, in the XXZ chain for 0 < W < 0.5 and L up to 400.The steady state has a uniform bulk spin-current profile, and this result agrees with expectations for non-integrable ergodic systems.Source: §3 · p. 53
  • For 0.5 < W < 2, NESS studies report subdiffusive j ∝L−γ with γ > 1, but increasing system size already shifts γ toward diffusion at W = 0.65.At W = 2, simulations reach only L ≤20, while slower relaxation makes the NESS increasingly difficult to obtain.Source: §3 · p. 53
  • Bulk-driven NESS temperature profiles provide a weak-coupling diagnostic: the ergodic regime follows δT ∝ϵ1/(2z), whereas stronger disorder yields a nonzero extrapolated profile as ϵ→0.The extracted exponent z rises abruptly beyond the diffusive value z = 2, while distinguishing an MBL phase requires ϵ →0 in the thermodynamic limit.Source: §3 · p. 54

B. Transport at large disorder strengths … 1. Quantum avalanches

At large disorder, transport slows dramatically, while rare ergodic bubbles provide a possible nonperturbative mechanism for destabilizing MBL. Avalanche propagation depends on localization length and geometry, but its relevance to disordered-spin-chain dynamics remains unresolved.

B. Transport at large disorder strengths

  • At W = 2, the fitted dynamical exponent decreases from z ≈ 3.8 to 3.1 across later time windows, indicating that observed subdiffusion may be transient.At W = 0.5, ΔS(t) ∝ t^1/z with z = 2, consistent with diffusion; simulations access only t < 100.Source: B. Transport at large disorder strengths · p. 54
  • For W < 3, the single-particle transport time grows exponentially, t1 ∝ e^aW, while transporting N = L/5 = 4 particles exceeds the Heisenberg time for L = 20 at W = 3.Increasing disorder imposes progressively stronger requirements on system size and evolution time.Source: B. Transport at large disorder strengths · pp. 54–55
  • The DC conductivity decreases exponentially, σ0 ∝ e−aW, beyond both the ergodic-regime boundary W_T(L) and the onset of MBL W∗(L), although extracting σ0 relies on potentially invalid broadening assumptions.These results show abrupt dynamical slowdown without determining the transport character or establishing an MBL phase.Source: B. Transport at large disorder strengths · p. 55

IX. MODELLING MBL TRANSITION

  • The modelling framework treats a weakly disordered rare region as an ergodic bubble coupled at its boundary to a strongly disordered MBL subsystem.The bubble follows ETH, the MBL subsystem is represented by exponentially localized LIOMs, and the coupling acts through interface operators.Source: §IX · p. 55, Figure 14 · p. 55, §1 · pp. 55–56

A. Rare thermal bubbles

  • Rare regions with locally weak disorder occur with probability pℓ = (W0/W)^ℓ = e−ℓlog(W/W0), yet arbitrarily large such regions exist with probability 1 as L → ∞.Their possible effects are therefore central to understanding disordered spin chains in the thermodynamic limit.Source: A. Rare thermal bubbles · p. 55

1. Quantum avalanches

  • Each thermalized neighboring spin doubles the bubble’s level density while coupling to more distant LIOMs decays as e−k/ξ, creating competition between bath growth and interaction decay.In one dimension, spreading in both directions multiplies the density of states by 4^k rather than 2^k.Source: §1 · p. 56
  • For ξ > ξ∗ = 1/ln 2, rare ergodic grains can thermalize surrounding MBL regions, whereas ξ < ξ∗ prevents avalanche propagation under the model’s assumptions.The prediction assumes the localization length remains unchanged when interactions are added.Source: §1 · p. 57
  • Avalanche predictions exclude an MBL phase in higher dimensions and for superexponentially decaying LIOMs, while numerical studies have not conclusively established their dynamical relevance.The quantum sun model offers a tractable interacting test bed with an ergodicity-breaking transition, a many-body mobility edge, and fading ergodicity, but its connection to disordered spin-chain dynamics remains open.Source: §1 · p. 57

2. Phenomenological investigations of MBL transition … 1. Thermalization of dilute impurities

The reviewed approaches probe the MBL transition through avalanche-based renormalization, planted thermal inclusions, open-system relaxation, and dilute impurities. Together, they expose finite-size drifts, support avalanche-consistent thermalization, and leave the thermodynamic-limit status and microscopic mechanism of MBL unresolved.

2. Phenomenological investigations of MBL transition

  • RG schemes model adjacent thermalizing and MBL blocks, whose matrix-element-driven growth ultimately classifies the system as ergodic or MBL.The avalanche mechanism is the leading proposal underlying these phenomenological renormalization approaches.Source: §2 · pp. 57–58
  • BKT-like scaling remains difficult to distinguish from power-law scaling: ν(L) grows indefinitely, yet ν(L = 10^6) ≈3 and exact diagonalization reaches only L ⪅20.Avalanche-based RG approaches yield a BKT universality class with an exponentially diverging length scale, while finite-size data can mimic power-law behavior.Source: §2 · p. 58

3. Probing avalanches with planted thermal inclusions

  • For planted thermal inclusions, ξ_d(t) grows faster than logarithmically and avalanches thermalize subsystem A for W_A ⪅W*(L), whereas growth saturates near ρ_B^-1 for W_A ⪆W*(L).The observed propagation is consistent with avalanches, but it does not exclude another microscopic mechanism with similar ξ_d(t).Source: §3 · p. 59
  • Planted inclusions support avalanche-consistent thermalization but retain ambiguities: rare regions, two length scales, and slow ergodic drifts prevent distinguishing an MBL phase from very slow thermalization.In one study, L_B = 6, W_B = 0.5, and W_A = 6 correspond to an estimated unplanted-region density of order 10^-6.Source: §3 · p. 59

4. Open system approach

  • The GKSL criterion requires the slowest relaxation rate Γ to decrease faster than 4^-L for avalanche stability, but gives W_avch(L = 14) ≈20, over five times the exact-diagonalization crossover W*(L = 14).The connection between this open-system construction and avalanches remains insufficiently clarified, and rare-region explanations are challenged by planted-inclusion results.Source: §4 · p. 60
  • Planted-inclusion dynamics show no significant drift in the disorder strengths marking qualitative ergodic-to-MBL changes, motivating further study of avalanche propagation and intermediate dynamical regimes.This conclusion concerns systems where inclusions may occur naturally only at sizes around L ≈10^6.Source: §4 · p. 60

B. Other approaches

  • The dilute-impurity approach reverses the avalanche perspective by testing whether strongly disordered sites thermalize within an otherwise ergodic chain, thereby probing finite-size effects.For V ≫1 at finite L, impurities generate LIOMs; the clean background uses W_0 = 1/4 and the impurity potential is V.Source: §1 · p. 60, B. Other approaches · p. 60

1. Thermalization of dilute impurities

  • Impurity studies find V* ∝L for single and finite-density impurities, while later gap-ratio crossings reflect Hilbert-space fragmentation rather than ergodicity breakdown; persistent linear drifts would imply thermodynamic-limit ergodicity at any V.The authors conclude that finite-density systems remain ergodic beyond L ∝V*, although specific Floquet systems can instead show sublinear drift.Source: §1 · p. 61

2. Models of many-body resonances … A. Other models of MBL

The review develops resonance-based probes of MBL and compares their predictions across disordered many-body models. These approaches identify finite-size resonance signatures, distinct disorder regimes, and model-dependent trends toward ergodicity or localization.

2. Models of many-body resonances

  • Resonance models assume LIOMs and show that proliferating many-body resonances destabilize MBL and trigger ergodicity, while reproducing several crossover features despite hosting an MBL phase.For the XXZ chain, the resonance length is estimated as 15 ⪅ λ ⪅ 50; the model reproduces linear W_T(L) drift, slow dynamics, apparent 1/ω spectral behavior, and exponential Thouless-time growth.Source: §2 · p. 61

3. Probing many-body resonances with exact diagonalization

  • Exact-diagonalization probes find enhanced end-to-end quantum mutual information in the intermediate ergodic–MBL crossover, where eigenstates are superpositions of limited numbers of substantially different spin configurations.At L = 16, W_swr ≈ 8.5 versus W*(L = 16) ≈ 3.5, while the crossing-point I_e-e is of order 10^-7; W_swr increases by approximately one-third from L = 11 to L = 16.Source: §3 · p. 62

4. Large-deviation approaches to resonances

  • Large-deviation sampling separates four disorder regimes, from RMT-like dynamics through rare resonances to exponentially suppressed escape, with only mild finite-size drift of W_c.For W_g < W < W_c, the initial state hybridizes with only O(1) distant states, whereas for W > W_c the probability of evolving away vanishes exponentially with n; the method also reproduces the Cayley-tree critical disorder accurately.Source: §4 · p. 63

X. POSSIBLE EXTENSIONS

  • Across extensions beyond the disordered XXZ chain, the review examines disordered Ising, Fermi-Hubbard, Bose-Hubbard, constrained, quasiperiodic, Floquet, long-range, higher-dimensional, and disorder-free systems.The supplied overview frames these systems as sharing some ergodicity-breaking features while also exhibiting distinguishing behaviors.Source: §X · p. 63

A. Other models of MBL

  • In the disordered TFIM, ETH-MBL crossover drifts resemble the XXZ case, while the simpler Fock-space graph has each basis state connected to exactly L single-spin-flip states.The crossing shifts sub-linearly with W*(L) ∝ 1/L, whereas the ergodic boundary follows W_T(L) ∝ L.Source: A. Other models of MBL · pp. 63–64
  • Disordered Ising chains support paramagnetic, spin-glass, and topological MBL signatures, but ergodicity is restored at crossovers between distinct MBL regimes.Numerical evidence places paramagnetic and spin-glass order at all energy densities, while the intervening ergodic regime is reported to be wide.Source: A. Other models of MBL · p. 64
  • The Fermi-Hubbard model exhibits distinct charge and spin dynamics, with studies asking whether ergodic spins can thermalize localized charges; a mobile impurity can instead induce gradual delocalization.At sufficiently large interaction and disorder, dynamics remain non-ergodic at accessible sizes and times, but intermediate regimes show subdiffusive impurity spreading.Source: A. Other models of MBL · pp. 64–65
  • Bose-Hubbard and constrained spin models display distinct finite-size behavior: the former has an unresolved inverted mobility edge, while PXP constraints produce stronger drifts toward asymptotic ergodicity.For PXP, W_T(L) and W*(L) shift linearly with L, with the trend becoming more pronounced for blockade radius α > 1; in Bose-Hubbard, persistence of the inverted mobility edge as L,t → ∞ remains open.Source: A. Other models of MBL · p. 65

B. Quasiperiodic potential … E. Long-range interactions

The review examines MBL signatures across quasiperiodic, alternative-disorder, Floquet, and long-range-interaction systems, finding modified ETH–MBL crossovers but persistent uncertainty about asymptotic localization. Floquet systems show especially suggestive finite-size trends, whereas long-range interactions may ultimately preclude an MBL phase.

B. Quasiperiodic potential

  • Quasiperiodic potentials produce an ETH–MBL crossover whose characteristic disorder depends strongly on the incommensurability factor η and lacks the large Griffiths regions found with random disorder.The crossover has been associated with W_cr ∼ln L rather than W_cr ∼L, with extrapolation giving W_cr ≈4 at L = 50.Source: B. Quasiperiodic potential · p. 66
  • In quasiperiodic systems, slow in-phase density oscillations and faster-than-exponential diffusion-constant decay suggest that localized dynamics may become visible at smaller sizes and times than with random disorder.Fock-space fluctuations, entropy measures, and level statistics also distinguish quasiperiodic from random systems.Source: B. Quasiperiodic potential · pp. 66–67

C. Other sources/types of disorder

  • Alternative disorder realizations broadly reproduce ETH–MBL crossover behavior: discrete disorder indicators collapse when plotted against on-site-potential variance, while correlated speckle disorder modifies crossover properties.Binary disorder permits thermodynamic-limit autocorrelation calculations, but accessible evolution only reaches t = 30, preventing reliable t →∞ extrapolation.Source: C. Other sources/types of disorder · p. 67
  • Random interactions can generate MBL-like eigenvalue, eigenvector, and dynamical signatures even when the noninteracting systems are delocalized Bloch-wave systems rather than Anderson localized.Finite-size drifts and dynamical features distinct from standard spin chains remain unresolved in related multifractal and interaction-disordered models.Source: C. Other sources/types of disorder · pp. 67–68

D. Floquet MBL

  • Floquet Kicked Ising results show a linear ergodic-boundary scaling at small sizes changing to sublinear growth, while the crossing drift weakens and extrapolations intersect near L ≈28.These trends provide premises for a stable Floquet MBL phase and involve weaker finite-size drifts than autonomous disordered chains.Source: D. Floquet MBL · p. 69

E. Long-range interactions

  • Resonance estimates constrain long-range MBL regimes by the interaction exponents and dimension, while random interaction coefficients enhance delocalization tendencies.These results apply to models with or without conservation of total spin, with the latter established for a more restricted class.Source: E. Long-range interactions · p. 69
  • Long-range interactions produce algebraically decaying LIOMs and algebraic entanglement growth, with S ∼t^γ and γ = 1/3 marking a crossover across broad interaction-range exponents.The same algebraic entanglement growth appears in infinite-range models, contrasting with logarithmic growth expected for short-range MBL.Source: E. Long-range interactions · p. 69
  • For power-law or slower interactions, avalanche arguments predict no asymptotic MBL phase because matrix-element decay cannot balance the growing density of states in an ergodic bath.Existing numerical results reach only L = 20 or larger systems at short tensor-network times, so further crossover-drift analysis is required.Source: E. Long-range interactions · pp. 69–70

F. Higher Dimensions … B. Outlook

This section examines MBL-like dynamics and related localization-protected phenomena across higher-dimensional, disorder-free, experimental, topological, Floquet, and discrete-time-crystal systems. It focuses on unresolved questions concerning the asymptotic MBL phase, its transition and microscopic mechanisms, persistent finite-size effects, and computational limits.

F. Higher Dimensions

  • Higher-dimensional systems show MBL-like behavior and geometry-dependent crossovers, but whether an MBL phase exists beyond one dimension remains unclear.Experiments and tensor-network studies observe strongly disordered particles retaining initial inhomogeneity or exhibiting very slow thermalization, while avalanche arguments preclude MBL beyond one dimension.Source: F. Higher Dimensions · p. 70

G. Ergodicity breaking without disorder

  • Disorder-free non-ergodic dynamics arise from approximate conservation laws, Hilbert-space fragmentation, and Stark localization, but their asymptotic ergodicity breaking remains an open problem.Gauge constraints, strong interactions, and dipole-moment conservation generate MBL-like phenomenology; pure tilted lattices can display behavior distinct from disordered Stark systems.Source: G. Ergodicity breaking without disorder · pp. 70–71

A. Early developments

  • Early experiments found transport at weak disorder and practically vanishing transport at strong disorder, consistent with slowed dynamics but not decisively establishing thermalization failure.In the disordered 3D Fermi-Hubbard model, the measured center-of-mass velocity decreased toward zero as disorder increased, while its relation to ergodicity remained unclear.Source: A. Early developments · p. 72

B. MBL regime in ultracold atomic systems

  • Ultracold-atom experiments established finite-time ETH–MBL crossovers, slow autocorrelation decay, entanglement growth, and signatures of avalanches, with coupling between tubes enhancing thermalization.Reported scales include tmax = 50 and 100 tunneling times in 1D Fermi-Hubbard experiments and L ≈500 sites with tmax = 300 tunneling times in a 2D Bose-Hubbard system.Source: B. MBL regime in ultracold atomic… · p. 72

C. MBL regime in other platforms

  • Other platforms reproduce MBL-regime signatures, including disorder-dependent crossovers in ion and superconducting-qubit systems and conductivity signatures in a disordered 2D electron system.The reviewed implementations also include finite-time quantum-avalanche expectations, energy-dependent crossover disorder, Fock-space dynamics, and Stark MBL.Source: C. MBL regime in other platforms · p. 73

XI. MBL IN EXPERIMENTS

D. Perspective on MBL experiments

  • Increasing experimental size and duration alone may reveal only further decay of observables, whereas probing microscopic mechanisms of ergodicity breaking could clarify the MBL phase.The review argues that identifying universal features of slow dynamics is important because finite coherence times make asymptotic trends experimentally inaccessible.Source: D. Perspective on MBL experiments · p. 73

A. Summary

  • Finite-size drifts toward ergodicity prevent reliable identification of the MBL phase, while slow dynamics, transport, and transition mechanisms remain key unresolved problems.Drifts may slow with system size or grow linearly, invalidating naive single-parameter scaling; the relevance of avalanches and many-body resonances requires further testing.Source: A. Summary · pp. 74–75
  • The MBL regime remains a key reference for ergodicity breaking and supports localization-protected quantum order, topological order, disordered Floquet phases, and discrete-time crystals.The enormous deep-MBL dynamical timescales make these phenomena practically stable, while their status as genuine stable dynamical phases remains tied to the MBL phase.Source: A. Summary · p. 75

B. Outlook

  • Experiments and theory have established many MBL-regime features, but the MBL phase and transition remain controversial despite their fundamental importance for robust, universal distinctions in statistical physics.The controversy concerns whether the observed regime corresponds to an asymptotically stable phase and transition.Source: B. Outlook · pp. 75–76
  • Microscopic mechanisms governing ergodicity and ergodicity breaking near the putative MBL transition remain important targets, regardless of the ETH-MBL crossover’s asymptotic fate.The situation is compared with unresolved debates in spin glasses and configurational glasses, where slow dynamics and competing physical scenarios likewise leave mechanisms unclear.Source: B. Outlook · p. 76
  • Classical numerical methods enabled current MBL understanding but remain constrained by interpretation drift, exponentially long evolution times, and the formal hardness of simulating many-body systems.More efficient GPU and TPU methods are an active challenge, while fault-tolerant quantum computers could open a new chapter; exponentially long times are likely needed for a final answer (Ehrenberg et al., 2022).Source: B. Outlook · p. 76

XIV. APPENDIX: DIFFERENT KINDS OF TRANSPORT AND COMPATIBLE EQUILIBRIA

Transport and equilibrium are distinct phenomena: systems can share an equilibrium distribution while exhibiting different transport, and ETH alone does not establish nonzero conductivity or diffusion. Subdiffusion can coexist with Gibbs equilibrium, whereas superdiffusion generally produces a non-Gibbsian stationary state.

Conceptual distinction

  • Transport and equilibrium can differ: distinct dynamical relaxation behaviors may converge to the same equilibrium distribution, so ETH does not imply nonzero conductivity or diffusion.The appendix distinguishes transport from equilibrium and gives classical counterexamples with anomalous transport and Gibbs equilibrium.Source: §XIV · p. 77

Subdiffusion

  • Subdiffusion from long-tailed waiting times remains compatible with Gibbs equilibrium, with γ = α/2 and mean-square displacement ⟨x2⟩ = tα.For α < 1, the waiting-time mean diverges; introducing a force still yields the stationary solution P ∝e−βV.Source: §XIV · p. 77

Superdiffusion

  • Broadly distributed jump intervals generate anomalous transport with γ > 1 or γ < 1, but the equilibrium distribution is not Gibbsian.The γ < 1 case corresponds to Lévy flights, while γ > 1 requires fine-tuning to suppress naturally dominant second-derivative terms.Source: §XIV · pp. 77–78

Implications for quantum systems

  • Thus, subdiffusion is compatible with Boltzmann–Gibbs equilibrium and likely some form of ETH, whereas superdiffusion leads to a non-Gibbsian equilibrium distribution.This framework is used to interpret transport observed in quantum spin chains at very small disorder.Source: §XIV · p. 78
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