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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

The review examines whether the finite-size, finite-time MBL regime becomes a genuine asymptotic phase without thermalization. It synthesizes numerical investigations and open questions, finding persistent drifts toward ergodicity alongside strong-disorder dynamical slowdown, leaving the phase's status unresolved.

  • Problem

    Whether the MBL regime survives as a phase without thermalization in the infinite-system-size and infinite-time limits remains unanswered.

  • Method

    The review analyzes numerical investigations and exact-diagonalization results on disordered many-body systems, emphasizing spectral properties, dynamics, and extrapolation challenges.

  • Results

    Persistent finite-size drifts toward the ETH regime prevent reliable single-parameter scaling and leave the ETH-MBL transition's existence and location undetermined.

  • Takeaways & Limitations

    Abrupt dynamical slowdown at increasing disorder suggests proximity to the MBL phase, while model-dependent results and slow dynamics keep its status open.

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

Statistical mechanics describes many-body systems through thermalization and ensemble averages, while quantum systems connect these ideas to ETH. Strong disorder can instead produce MBL, but its asymptotic status remains unresolved.

  • Statistical mechanics describes long-time behavior through statistical ensembles, allowing time averages of observables to be calculated as ensemble averages.
  • ETH links quantum thermalization to equilibrium behavior of individual many-body eigenstates in isolated non-integrable systems.
  • Strong disorder and interactions can inhibit thermalization, producing MBL systems that retain initial-state information through local integrals of motion.
  • The existence of an MBL transition matters for non-equilibrium phenomena, diffusive transport, and the applicability of statistical physics, but numerical interpretations remain inconclusive.
  • Defining an MBL phase requires assessing thermalization in the joint limits t →∞ and L →∞, which is difficult because relaxation times grow with system size and slowing dynamics.

D. Modeling and phenomenology of MBL

MBL exhibits strong-disorder non-ergodic behavior, but whether this finite-size regime constitutes an asymptotic MBL phase remains unresolved. Numerical evidence shows size-dependent spectral drifts and persistent dynamical decay that complicate identifying the ETH–MBL transition.

  • Strongly disordered XXZ chains display slow dynamics and low-entanglement eigenstates interpreted as signatures of MBL.
  • LIOMs explain initial-state memory, absent transport, Poisson eigenvalue statistics, area-law eigenstates, and slow entanglement spreading in the MBL phase.
  • Constructing LIOMs remains difficult because existing perturbative and numerical methods are approximate or limited to small systems, while rigorous constructions remain unsettled.
  • Quantum sun models provide a reference where numerical simulations accurately determine ergodicity-breaking transitions, but their relevance to disordered spin chains and higher dimensions remains unclear.
  • Finite-size spectral crossings in one-dimensional disordered systems drift toward larger disorder, making the critical disorder and even the existence of an MBL phase difficult to identify.
  • At stronger disorder, observable decay slows substantially, yet its continued strengthening with system size leaves the infinite-size, infinite-time fate undecided.

C. Remarks on finite-size scaling

The review argues that naive single-parameter scaling cannot capture persistent finite-size drifts at the ETH-MBL crossover. More elaborate scaling forms improve flexibility but remain difficult to distinguish and constrain with available system sizes.

  • Finite-size scaling is used to investigate whether the ETH-MBL crossover represents a true phase transition and to understand its behavior.A successful analysis could address both the existence of an MBL phase and the crossover mechanism.
  • Single parameter scaling: Single-parameter scaling fails to reproduce the significant drift of crossing points observed near the ETH-MBL crossover.Under continuous, monotonic scaling functions, the crossing point is forced to equal the critical disorder, contrary to the observed drift.
  • Single parameter scaling: Visually appealing data collapses can still produce unreliable critical exponents when crossing-point drift is disregarded.Restricting analyses to the largest available sizes may conceal the crossover's system-size dependence.
  • Sub-leading corrections to the scaling: BKT and power-law scaling scenarios can be difficult to distinguish because their finite-size transition-point drifts differ only gradually.A toy model was initially fitted with ν ≈3 under power-law scaling before refined analysis favored the BKT scenario.
  • Sub-leading corrections to the scaling: Multi-parameter and unconventional scaling forms account for drifts but have limited predictive power when system sizes are restricted.The available many-body results cannot reliably determine critical exponents when scaling forms contain multiple free parameters.
  • Unconventional scaling limits: Unconventional scaling can yield transition points that grow with system size, while other limits retain a finite critical disorder.Examples include linear drift in the XXZ chain and constrained spin chains, alongside alternative thermodynamic-limit constructions.
  • Unconventional scaling limits: More complex scaling analyses are required when finite-size drifts invalidate a simple zero of the beta function.Anderson-model examples provide a cautionary comparison for interpreting MBL crossover data.

D. Conclusion

The review concludes that finite-size and time drifts prevent a decisive distinction between slow dynamics and true MBL in the asymptotic limit. It surveys numerical approaches while emphasizing both their reach and their limitations.

  • Conclusion: Finite-size and time-dependent drifts toward ETH leave the existence and asymptotic fate of the MBL phase unresolved.The central difficulty is distinguishing slow dynamics from complete dynamical arrest as L,t →∞.
  • Conclusion: Single-parameter scaling is insufficient for ETH-MBL crossover data, requiring irrelevant variables or unconventional scaling functions.The review presents these corrections as necessary to capture system-size dependence.
  • Conclusion: Anderson models illustrate distinct critical-scaling behaviors, including finite-size delocalization that shrinks thermodynamically and strong crossover drifts.These cases motivate caution when interpreting disordered many-body systems.
  • Numerical methods: Shift-and-invert methods target eigenvalues near a chosen energy and extend exact middle-spectrum calculations to L ≤26.The transformation maps target-near eigenvalues to spectrum edges, where Lanczos converges efficiently.
  • Numerical methods: Chebyshev and Krylov evolution enable numerically exact dynamics beyond full exact diagonalization, reaching Hilbert-space dimensions of approximately 10^9.Chebyshev evolution can be a couple of times faster than Suzuki-Trotter schemes.
  • Numerical methods: Exact state-vector methods remain limited by exponential Hilbert-space growth and cannot address systems of roughly 100 sites.This restricts direct comparison with experimentally relevant system sizes.
  • Numerical methods: Light-cone renormalization-group methods study one-dimensional systems directly at L →∞ by restricting operator evolution to an effective light cone.The approach relies on Lieb-Robinson bounds and requires translational invariance supplied through discrete disorder models.
  • Numerical methods: Tensor-network approaches can target putative area-law eigenstates but may miss subtle effects signaling thermalization.Their bias toward less-entangled states is a stated limitation.

D. Outlook

Numerical studies reveal persistent finite-size ambiguities in distinguishing an asymptotic MBL phase from eventual ergodicity. Gap-ratio and spectral-form-factor analyses expose incompatible scaling trends, while exponentially slowed dynamics preserve the possibility of MBL without establishing it.

  • D. Outlook: Exact diagonalization gives numerically exact ETH–MBL crossover results, but exponentially growing Hilbert spaces restrict accessible system sizes.Tensor-network methods reach larger systems but require careful convergence monitoring.
  • 1. Gap ratio: For weak disorder W ⪅6, gap ratios trend toward GOE, whereas for strong disorder W ⪆10 they decrease with system size toward Poisson statistics.The intermediate regime exhibits a significant drift of the crossing point of r(W) curves.
  • 1. Gap ratio: The crossing and rescaling trends imply incompatible finite-size scalings, with W T (L) < W ∗(L), preventing a straightforward thermodynamic-limit interpretation.The analysis cannot unambiguously distinguish a finite-disorder MBL transition from eventual ergodicity at every disorder strength.
  • 1. Gap ratio: Distinguishing the competing scenarios may require system sizes near L0 ≈50, beyond present-day exact diagonalization capabilities for middle-spectrum eigenstates.The finite-size data alone cannot entirely exclude either scenario.
  • 2. Thouless time: The spectral form factor reaches GOE predictions only after the Thouless time, while in the strong-disorder regime it slowly approaches the Poisson result K_PS(τ) = 1 with increasing system size.Convergence to Poisson statistics for L →∞ at W > Wc would imply an MBL phase.
  • 2. Thouless time: Thouless time grows approximately exponentially with disorder, producing dramatic dynamical slowdown that obscures the distinction between slow thermalization and arrested dynamics.Because the scaling may fail when tTh approaches tH or at asymptotically large L and W, it does not by itself rule out MBL.

3. Other measures of spectral statistics

Spectral statistics diagnose the ETH–MBL crossover through level correlations, spacing ratios, and long-range measures, but their crossing points and timescales drift with system size, complicating phase-boundary identification.

  • Level-spacing distributions evolve with system size across the ETH–MBL crossover, motivating the simpler average gap ratio.
  • Higher-order spacing ratios extend level-statistics analysis to larger energy scales and exhibit linear system-size drifts at the ETH–MBL crossover.
  • Level statistics provide a direct exact-diagonalization diagnostic: random-matrix behavior signals ETH, whereas Poisson flow indicates broken ergodicity.
  • The average gap ratio probes single-level spectral scales and the implicit t→∞ limit, yet its ETH–MBL crossover shifts significantly with system size.
  • The spectral form factor defines a Thouless time whose scaling remains ambiguous across the crossover, while increasing disorder produces an approximately exponential increase.
  • Eigenstate multifractality distinguishes regimes, with Dq = 1 in ETH and 0 < Dq < 1 in MBL, but Dq crossing points drift toward larger W as L increases.

2. Quantum information measures

Quantum-information and dynamical spectral measures characterize ergodicity breaking through entanglement, mutual information, spectral decay, and memory retention. These observables show ETH-to-MBL changes but retain finite-size ambiguities and disorder-dependent dynamical slowdowns.

  • Entanglement measures: Quantum mutual information is nearly separation-independent for ETH eigenstates, whereas MBL eigenstates exhibit area-law entanglement and reduced correlations.
  • Entanglement measures: Rescaled entanglement entropy s approaches 1 in the ergodic regime and decreases toward 0 in the MBL regime as system size increases.
  • Finite-size effects: Crossing-point drifts depend on the observable: r and s drift more than Dq, while QMI can drift more weakly than entanglement entropy.
  • Entanglement measures: The standard deviation of entanglement entropy grows superlinearly, σS ∝ L^α with α > 1, across the available crossover sizes.
  • Dynamical measures: The spectral function develops a plateau between ωH and ωTh in the ergodic regime, with a high-frequency tail f2(ω) ∝ ω^-1/2 indicating diffusive decay.
  • Dynamical measures: Increasing disorder narrows the spectral plateau, lowers ωTh/ωH, and increases the decay exponent, reflecting slowed dynamics; the plateau disappears in the MBL regime.
  • Dynamical measures: The stiffness C0 decays exponentially with L in ETH but remains approximately L-independent in MBL, without settling the L→∞ fate.

2. Sensitivity to local perturbations

Local perturbations reveal distinct spectral responses across ergodic and MBL regimes, while fidelity-susceptibility and matrix-element studies expose persistent ambiguity about the ETH-MBL transition. Dynamical results likewise show disorder-dependent slowing and finite-size trends that complicate extrapolation to infinite size and time.

  • Spectral response to local perturbations: In the ergodic regime, eigenvalues undergo irregular motion with multiple avoided crossings, whereas the MBL regime shows more regular motion and localized constants of motion.The avoided crossings in the MBL regime are interpreted as resonances between LIOM configurations, becoming more extensive as disorder decreases toward the crossover.
  • Spectral response to local perturbations: Level velocities and curvatures locate the ETH-MBL crossover, but deviations from random-matrix predictions appear even deep in the ETH regime.These quantities are derived from eigenvalue responses to the perturbing operator ∂λH.
  • Fidelity susceptibility: χtyp ∝ 2^L in the ETH regime, while its rescaled form crosses near W* ≈ 4.5 and drifts toward larger disorder with increasing L.The ETH scaling is reported for W < WT(L) ≈ 1.5–2.1 and L ∈ [12,18].
  • Fidelity susceptibility: The fidelity-susceptibility distribution challenges assumptions underlying mathematical arguments for MBL stability because matrix elements and energy denominators are statistically correlated.The observed distribution is described as reminiscent of emergent level attraction, while the relevant independence assumption is reported to be invalid.
  • Matrix elements and dynamics: Matrix elements of local observables link spectral properties to dynamics, yet their observed trends do not permit unambiguous conclusions about the ETH-MBL crossover as L → ∞.The review emphasizes that this ambiguity appears across spectral, eigenstate, and dynamical analyses.
  • Matrix elements and dynamics: In interacting chains, density autocorrelations decay persistently but increasingly slowly with disorder, and extrapolations from finite L and t remain uncertain.At W = 4 and ∆ = 1, decay continues approximately to tH ∝ e^L; at W = 5, decay is slower than logarithmic in the displayed finite systems.

2. Assessing the breakdown of ergodicity

Time-evolution studies reveal extremely slow dynamics deep in the MBL regime, but finite-size and finite-time effects prevent deciding whether thermalization ultimately persists or stops. Autocorrelation, transport, and entanglement results consistently show trends toward slow thermalization alongside an abrupt disorder-induced slowdown.

  • Persistent slow dynamics: At W = 8, the density autocorrelation C(t) decays slowly and is fitted by C(t) ∝t^-β with β ≈10^-3.The decay is observed over times of order 10^3 tunneling times using TDVP.
  • Persistent slow dynamics: Across W ∈[1, 10], the decay exponent β decreases exponentially with disorder strength, β ∝e^-γW.The trend appears across several computational approaches and literature studies of the disordered XXZ chain.
  • Asymptotic uncertainty: Present numerical results cannot distinguish whether C(t) remains algebraic and thermalizes the system or crosses over to slower decay and an MBL phase.The competing scenarios differ in whether the critical disorder converges to a finite value or diverges with system size.
  • Asymptotic uncertainty: If the W = 8 exponent β ≈10^-3 persisted, C(t) would require t ≈10^1000 tunneling times to reach 10% of its initial value.This would make the dynamics effectively arrested for practical purposes, without resolving the asymptotic fate.
  • Transport and scaling: At increasing system size, the dynamical exponent z decreases despite an exponential spatial-correlation profile, consistently with trends toward thermalization.These finite-time and finite-size effects become more pronounced at stronger disorder.
  • Entanglement growth: For W = 10, entanglement growth is better described by S(t) ∼t^γ with γ ≈0.08 over a broad interval, while logarithmic growth fits a shorter interval.The distinction between algebraic and logarithmic growth remains unresolved at longer times, so entanglement data do not establish an MBL phase.

2. Symmetry resolved entanglement: number entropy

Symmetry-resolved entanglement separates configurational entanglement from number fluctuations across a subsystem’s conserved-spin sectors. Number entropy displays ultraslow growth whose interpretation remains contested, while related analyses connect slow dynamics to broad relaxation-time distributions and weak perturbations of Anderson localization.

  • Symmetry resolution: The XXZ chain’s U(1) symmetry makes the reduced density matrix block diagonal in subsystem spin sectors, enabling symmetry-resolved entanglement.Projectors Π_s select sectors indexed by subsystem spin eigenvalues, with probabilities p_s determined by the reduced density matrix.
  • Symmetry resolution: Number entropy S_n measures the probability distribution over the subsystem’s conserved-spin sectors, distinct from configurational entanglement within sectors.This decomposition provides a separate probe of particle or spin transport across the bipartition.
  • Number entropy dynamics: At W = 10, TDVP results for L = 50 show number entropy growth accurately captured by S_n(t) ∝ln ln(t).The same study finds entanglement entropy better fit by a power law over a broader time interval.
  • Number entropy dynamics: The double-logarithmic number-entropy growth has been interpreted both as persistent slow particle transport and as a transient single-particle boundary fluctuation.Available numerical results can accommodate both interpretations and require larger sizes and longer times for resolution.
  • Number entropy dynamics: A LIOM circuit in the MBL phase also exhibits S_n(t) ∝ln ln(t), persisting to L = 28 and times exponential in L, though larger-size saturation is not excluded.This supports, but does not establish, ultraslow number-entropy growth as an intrinsic MBL feature.
  • Open status: Slow local-correlator decay and gradual entanglement growth persist across the largest sizes and times currently accessible numerically, leaving the ETH-MBL transition unresolved.The time-domain evidence demonstrates a robust slow-dynamics regime but not an asymptotic phase boundary.
  • Proximity to Anderson insulator: Interactions destabilize Anderson LIOMs through broad relaxation-time distributions, producing persistent local-correlator decay and nontrivial dynamics in the MBL regime.The relaxation-time distribution follows a power law over a wide range, while a fraction of LIOMs remains stable beyond the Heisenberg time.
  • Proximity to Anderson insulator: A weak but non-vanishing interaction perturbation yields slow dynamics toward equilibrium, whereas rescaling the perturbation or adding strong pair hopping restores ergodicity at any disorder strength.These results motivate viewing strongly disordered chains as weakly perturbed Anderson insulators.

2. Stretched exponential decays

Near the ergodic boundary, local autocorrelations can show stretched-exponential decay, but this prediction breaks down at accessible sizes and disorder strengths. Entanglement-based rescaling reveals universal slow dynamics without visible MBL signatures, while operator-growth results likewise find no convincing MBL evidence at probed scales.

  • Stretched exponential decays: Near W ≈ W_T(L), local autocorrelations decay faster than the power law at larger disorder and fit a stretched exponential with κ < 1.The characteristic time scale τ increases with W.
  • Stretched exponential decays: Many-body resonances are proposed as an intermediate step toward local equilibration, after which hydrodynamic power-law decay is expected to dominate.Power-law decay is also observed in quasiperiodic systems lacking rare regions.
  • Stretched exponential decays: The stretched-exponential prediction breaks down for L = 20 already at W = 3, where C(t) is also consistent with power-law or logarithmic decay.The available time interval prevents strong conclusions about the functional form.
  • Internal clock of many-body dynamics: For W < 8, disorder-averaged C(t) and its fluctuation collapse onto a system-size-dependent master curve when plotted against entanglement entropy S(t).This identifies S(t) as an internal clock and yields no visible MBL signatures in the studied sizes and times.
  • Internal clock of many-body dynamics: An MBL phase would require a separate master-curve branch with indefinitely growing fictitious time and non-decaying autocorrelations, but none appears for t < 10^3.Evidence for such a phase would need larger times and system sizes.
  • Operator spreading: Nested-commutator growth remains increasing at W = 25 through n = 16, probing operators over L = 33 sites and providing no convincing MBL signature there.Interacting operators accelerate their growth relative to non-interacting ones after n ≲ 8.

A. Transport in the ergodic regime

Transport is clearly diffusive deep in the ergodic regime, but finite-size and finite-time effects make the crossover at larger disorder difficult to classify. Conductivity and dynamical studies consistently show slowing dynamics and trends toward diffusion or unresolved transport behavior.

  • A. Transport in the ergodic regime: The diffusion constant decreases exponentially with disorder, D_0 ∝ e^-aW, and deviations from RMT become observable when it approaches the many-body level spacing.This occurs at W ∝ L, similarly to the finite-size ergodic threshold.
  • A. Transport in the ergodic regime: At W = 1.5, the maximal dynamical exponent decreases from z ≈ 4.2 at L = 16 to z ≈ 2.95 at L = 32, drifting toward the diffusive limit z = 2.The values are extracted after the initial transient and before late-time saturation.
  • A. Transport in the ergodic regime: Anderson-model results show that apparent subdiffusion can drift toward diffusion only at larger sizes, with L = 80 suggesting α ≈ 0.75 while L = 120 reveals α → 1.The required scales exceed present-day full exact diagonalization for the 3D model.
  • A. Transport in the ergodic regime: Deep in the ergodic XXZ regime, systems up to L = 400 show diffusive spin transport with j ∝ L^-1.The transport exponent satisfies γ = z − 1, so diffusion corresponds to z = 2.
  • A. Transport in the ergodic regime: For 0.5 < W < 2, reported subdiffusive current scaling j ∝ L^-γ with γ > 1 remains inconclusive because increasing system size trends toward γ = 1.Spin transport may differ from energy transport in disordered models.
  • A. Transport in the ergodic regime: Increasing disorder causes abrupt growth of the dynamical exponent beyond z = 2, while the resulting slowdown hinders reliable transport classification.The same barrier affects NESS, unitary dynamics, and conductivity analyses.

B. Transport at large disorder strengths

At large disorder, transport slows dramatically and numerical access becomes insufficient to determine its asymptotic character or establish an MBL phase. Avalanche theory offers a mechanism for thermalization by rare ergodic regions, but its relevance and assumptions remain unsettled.

  • B. Transport at large disorder strengths: At W = 0.5, domain-wall spreading follows diffusion with z = 2, whereas at W = 2 fitted exponents decrease from z ≈ 3.8 to z ≈ 3.1 over successive time windows.The simulation accesses only relatively short times, t < 100.
  • B. Transport at large disorder strengths: The single-particle crossing time grows exponentially with disorder, t_1 ∝ e^aW for W < 3, and transport of O(L) particles may exceed the Heisenberg time.The increase becomes faster at larger disorder strengths.
  • B. Transport at large disorder strengths: The DC conductivity continues its exponential decrease, σ_0 ∝ e^-aW, beyond the onset of the putative MBL regime.This behavior is identified as another manifestation of abrupt dynamical slowdown.
  • B. Transport at large disorder strengths: Persistent slow dynamics prevent definite conclusions about either the MBL phase or transport at stronger disorder.Numerical and experimental studies do not determine the transport character in strongly disordered many-body systems.
  • Quantum avalanches: Rare weak-disorder regions can form ergodic bubbles whose coupling to MBL regions may trigger avalanches that thermalize neighboring subsystems and destabilize LIOMs.If ξ < ξ*, the avalanche does not propagate throughout the lattice, predicting absence of thermalization within the scenario.
  • Quantum avalanches: The avalanche mechanism predicts no MBL phase in higher dimensions and relies on balancing exponentially decaying interaction matrix elements against exponentially shrinking bubble level spacings.Its relevance to disordered spin-chain dynamics remains to be determined.
  • Quantum avalanches: The quantum sun model numerically realizes an ergodicity-breaking transition with only mild finite-size drift in the gap-ratio crossing point.It provides a test bed for phenomena unclear in disordered XXZ chains.

2. Phenomenological investigations of MBL transition

Avalanche-based RG approaches model MBL transitions through growing thermal and localized blocks, while planted-inclusion studies probe whether ergodic bubbles thermalize their surroundings. Numerical results reveal strong finite-size dependence, but abrupt dynamical slowdowns and disorder-dependent propagation support proximity to an MBL phase.

  • Avalanche RG: Avalanche RG divides disordered chains into thermalizing and MBL blocks whose growth determines whether the system is globally ergodic or localized.Thermal blocks can absorb neighboring MBL blocks when matrix elements exceed their level spacing.
  • Avalanche RG: Avalanche-based RG approaches yield a BKT universality class, whose exponentially divergent length scale makes numerical discrimination from power-law scaling difficult.At L = 10^6, the effective exponent can still be approximately ν(L = 10^6) ≈3.
  • Planted thermal inclusions: Thermal bubbles in weak-disorder regions can seed avalanches, but propagation stops near W ∗(L), even when the planted inclusion would normally require much larger chains.For example, a bubble with LB = 6 and WB = 0.5 was studied inside a subsystem with WA = 6.
  • Planted thermal inclusions: Thermal inclusions accelerate autocorrelation decay, but the rescaling t →te−aW collapses results only with periodic driving and not when the coupling remains continuously present.The reported rescaling uses a = 2.1.
  • Stability criterion: MBL stability against avalanches in the open-system criterion requires the slowest decay rate Γ to decrease with L faster than 4−L.This criterion comes from analyzing a disordered chain coupled to a large thermal inclusion using a GKSL equation.
  • Finite-size drifts: Despite planted inclusions appearing only around L ≈10^6, simulations found no significant drift in the disorder strengths separating qualitative ergodic and MBL dynamics.The review therefore calls for further study of microscopic avalanche propagation and ergodicity emergence.
  • Finite-size drifts: For finite impurity densities, ergodicity indicators and impurity thresholds can drift linearly with system size, while crossing points need not mark ergodicity breakdown.The LIOMs are already destabilized at V = V ∗∝L, before the crossing-point behavior becomes informative.

2. Models of many-body resonances

Resonance-based models describe MBL destabilization through sparse, system-spanning hybridizations and reproduce several finite-size and dynamical signatures of the ETH-MBL crossover. Large-deviation analyses further distinguish regimes ranging from random-matrix behavior to exponentially suppressed spreading, while other models show that apparent crossing points may not identify the transition.

  • Resonance models: Many-body resonances hybridize weakly entangled product states that differ across extensively many regions, producing large local-correlation oscillations and potentially destabilizing MBL.Resonance proliferation is associated with the onset of ergodicity.
  • Resonance models: The resonance model uses ξ for the typical resonance range and λ for resonance density, with perturbative arguments predicting ξ divergence at the MBL transition.For the disordered XXZ chain, λ is estimated as 15 ⪅λ ⪅50.
  • Resonance models: For 15 ⪅λ ⪅50, the resonance model reproduces linear crossover drifts, slow dynamics, apparent 1/ω spectral behavior, and exponentially increasing Thouless times.These features arise in a model that, by construction, contains an MBL phase.
  • Exact-diagonalization probes: The system-wide resonance indicator crosses at W ∗(L = 16) ≈8.5, compared with W ∗(L = 16) ≈3.5 from standard level statistics.At its crossing, the resonance maximum is of order 10−7, yet the resonances span the entire chain.
  • Exact-diagonalization probes: Combining characteristic disorder scales led to the conclusion that an MBL phase, if present in the disordered XXZ chain, requires W ⪆20.The review identifies unresolved questions about relating open-system and isolated-chain indicators.
  • Large-deviation approaches: A free-energy analysis of T0(β) distinguishes four disorder regimes, from random-matrix-like spreading to exponentially vanishing escape from the initial state.In the intermediate regime Wg < W < Wc, only O(1) distant states hybridize with the initial state.
  • Large-deviation approaches: The critical disorder Wc shows only mild finite-size drift, and the same method reproduces the known Anderson-model threshold on a Cayley tree accurately.This supports the use of the large-deviation approach as a reference for resonance analysis.
  • Other models: In the PXP model, stronger blockade constraints permit larger exact-diagonalization sizes, while W T(L) ∝L and W ∗(L) ∝L become more pronounced for α > 1.These trends suggest the PXP models remain ergodic asymptotically.

B. Quasiperiodic potential

Quasiperiodic and other nonstandard disorder models reproduce aspects of MBL phenomenology while changing the structure of rare regions, transport, and finite-size extrapolation. These settings broaden the comparison space but leave several asymptotic questions unresolved.

  • Quasiperiodic potential: Quasiperiodic potentials generate an ETH-MBL crossover as their strength W increases and are routinely realizable in ultracold-atom experiments.The potential includes amplitude W, incommensurability η, and a random phase φ.
  • Quasiperiodic potential: Strong spatial correlations in quasiperiodic disorder eliminate large Griffiths regions of anomalously weak or strong disorder.This distinguishes quasiperiodic systems from systems with independent random onsite potentials.
  • Quasiperiodic potential: The diffusion constant in the quasiperiodic XXZ chain decays faster than exponentially, while Fock-space observables distinguish quasiperiodic and random systems through sample-to-sample fluctuations.Similar distinctions appear in entropy measures and level statistics.
  • Alternative disorder models: Binary disorder can be averaged over all 2^L realizations in a single enlarged-system evolution, enabling autocorrelation calculations directly at L →∞.The accessible evolution time is only t = 30, preventing reliable extrapolation to t →∞.
  • Interaction-induced localization: Random interactions can produce localization-like many-body behavior even when the noninteracting systems are delocalized Bloch-wave problems.Related studies include spinful and spinless fermion models with random interactions.
  • Alternative disorder models: Weak links require dynamical bounds that distinguish all-site guarantees from bounds valid only on subsets of the chain.Standard Lieb-Robinson bounds do not capture the full complexity induced by anomalously small tunneling bonds.
  • Floquet and long-range models: In the Kicked Ising model, the crossing point drifts sublinearly and the boundary of the ergodic regime drifts approximately linearly with L, suggesting a possible stable MBL phase.The review relates weaker Floquet drifts partly to the absence of energy conservation.
  • Floquet and long-range models: Effectively long-range models generated by eliminating dynamical gauge fields display MBL-like phenomenology, but their asymptotic ergodicity breakdown remains an open question.Long-range interactions can also produce slow entanglement growth despite eventual thermalization in some settings.

F. Higher Dimensions

Higher-dimensional and constrained many-body systems exhibit geometry- and model-dependent ergodicity-breaking behavior, but available evidence remains insufficient to settle their asymptotic phases. Across models, finite-size drifts, limited evolution times, and persistent thermalization tendencies constrain conclusions about MBL.

  • F. Higher Dimensions: Studies of higher-dimensional ergodicity breaking are much rarer than one-dimensional studies, with early two-dimensional experiments probing disorder-dependent particle spreading.Small-disorder systems filled the lattice almost uniformly after particles were released.
  • F. Higher Dimensions: In two-dimensional spinless fermion systems, the crossover to MBL depends on lattice geometry and the number of directly connected nearest neighbors.This dependence is consistent with a possible approximate mean-field description.
  • F. Higher Dimensions: Gauge-field models that become effectively long-ranged after imposing Gauss's law exhibit MBL-like phenomenology, but their L,t →∞ ergodicity breakdown remains unresolved.The review identifies these models as a direction for further research.
  • Asymptotic limits: Strongly disordered systems may fail to thermalize over experimentally relevant times t < tmax, motivating searches for universal features of slow dynamics.Finite-time nonthermalization does not by itself establish an asymptotic MBL phase.
  • Asymptotic limits: Exact diagonalization reaches only L ≈20, while larger time-evolution systems face persistent flow toward thermalization and finite experimental coherence times.These constraints make simultaneous extrapolation to L,t →∞ impractical.
  • Asymptotic limits: Finite-size drifts may slow or grow linearly with L, preventing reliable single-parameter scaling and leaving the transition disorder and even the MBL phase status undetermined.The review contrasts potentially slowing drifts with linear drifts suggesting eventual thermalization at any W.
  • Open questions: The MBL regime is well characterized numerically, but determining which of its features survive asymptotically remains an essential challenge.The review highlights avalanche mechanisms and links between resonance probes and standard ergodicity indicators as priorities.
  • Open questions: Classical-computing methods using CPUs dominate current MBL studies, while GPU- and TPU-based approaches are identified as an important efficiency challenge.The review frames computational development as part of the continuing investigation of many-body localization.

XIV. APPENDIX: DIFFERENT KINDS OF TRANSPORT AND COMPATIBLE EQUILIBRIA

Transport and equilibrium are distinct but related aspects of localization, and anomalous transport can coexist with different equilibrium distributions. Continuous-time random-walk examples show that subdiffusion can retain Gibbs equilibrium, whereas superdiffusion generally produces non-Gibbsian equilibrium.

  • Transport and equilibrium: Transport describes dynamical relaxation, while equilibrium concerns distributions or eigenstate properties; these need not determine one another.The passage explicitly asks whether different transport behaviors can share the same equilibrium state and answers yes in principle.
  • Transport and equilibrium: ETH in eigenstates alone does not imply a non-zero conductivity or diffusion coefficient.Classical dynamical counterexamples demonstrate anomalous transport alongside convergence to the Gibbs distribution.
  • Continuous-time random walks: Continuous-time random walks with independent waiting-time and spatial-jump distributions provide examples of anomalous transport.The transport exponent can arise from broad waiting-time distributions or broadly distributed jumping intervals.
  • Compatible equilibria: Subdiffusion is compatible with the Boltzmann-Gibbs distribution, including when the dynamics is described by a fractional time derivative.With an applied force, the stationary solution remains Gibbsian, with β related to D and µ through the Einstein relation.
  • Compatible equilibria: Superdiffusion leads to a non-Gibbsian equilibrium distribution, while the corresponding γ > 1 case requires fine-tuning to remove naturally dominant second-spatial-derivative terms.The γ < 1 case corresponds to Lévy flights and is described as more natural than γ > 1.
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