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Geometry-resolved avalanche criterion in disordered spin systems: transverse connectivity corrections and asymptotic recovery of the flat-front law

Hikaru Wakaura, Taiki Tanimae

Published Jul 20, 2026• 0 citations
Publisher

Abstract

Abstract Isolated quantum many-body systems can fail to thermalize in the presence of strong disorder, a phenomenon known as many-body localization (MBL). Whether this breakdown of statistical mechanics survives in the thermodynamic limit is believed to be decided by quantum avalanches: rare thermal inclusions that grow by consuming their localized surroundings, a mechanism that renders MBL unstable in more than one dimension. Quantitative tests of the avalanche criterion have so far been confined to strictly one-dimensional chains, leaving the geometry dependence at the heart of the higher-dimensional argument unmeasured. Here we show, using a bath-coupled relaxation-rate diagnostic validated against exact Lindblad dynamics, that the avalanche stability criterion acquires systematic and lawful geometry corrections as a disordered XXZ system is widened from a chain to two-, three-, and four-leg strips and to quasi-three-dimensional tubes. In contrast to the flat-front expectation, in which geometry enters only through the cross-sectional area, we find that the measured decay slope of the slowest relaxation rate decreases with transverse connectivity: the slope deficit is organized, to leading approximation, by the number of transverse bonds per column, and geometries containing closed transverse loops retain their deficits up to the strongest disorder accessible. At the same time, all deficits vanish as disorder grows, so that the area-only avalanche criterion is recovered asymptotically. These results promote the avalanche instability argument from a scaling estimate to a measured, geometry-resolved law, explain why apparent critical disorder grows with system width more slowly than analytic predictions, and provide the quantitative framework needed to interpret the system-size drift of the MBL crossover observed in two-dimensional quantum simulators.

DOI
10.21203/rs.3.rs-10389011/v1

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Abstract Isolated quantum many-body systems can fail to thermalize in the presence of strong disorder, a phenomenon known as many-body localization (MBL). Whether this breakdown of statistical mechanics survives in the thermodynamic limit is believed to be decided by quantum avalanches: rare thermal inclusions that grow by consuming their localized surroundings, a mechanism that renders MBL unstable in more than one dimension. Quantitative tests of the avalanche criterion have so far been confined to strictly one-dimensional chains, leaving the geometry dependence at the heart of the higher-dimensional argument unmeasured. Here we show, using a bath-coupled relaxation-rate diagnostic validated against exact Lindblad dynamics, that the avalanche stability criterion acquires systematic and lawful geometry corrections as a disordered XXZ system is widened from a chain to two-, three-, and four-leg strips and to quasi-three-dimensional tubes. In contrast to the flat-front expectation, in which geometry enters only through the cross-sectional area, we find that the measured decay slope of the slowest relaxation rate decreases with transverse connectivity: the slope deficit is organized, to leading approximation, by the number of transverse bonds per column, and geometries containing closed transverse loops retain their deficits up to the strongest disorder accessible. At the same time, all deficits vanish as disorder grows, so that the area-only avalanche criterion is recovered asymptotically. These results promote the avalanche instability argument from a scaling estimate to a measured, geometry-resolved law, explain why apparent critical disorder grows with system width more slowly than analytic predictions, and provide the quantitative framework needed to interpret the system-size drift of the MBL crossover observed in two-dimensional quantum simulators.
We investigate the stability of the many-body localized phase against quantum avalanche instabilities in a one-dimensional Heisenberg spin chain with long-range power-law interactions (V∝r−αV\propto r^{-α}V∝r−α). By combining exact diagonalization of static properties with Lindblad master equation simulations of open-system dynamics, we systematically map the interplay between interaction range and disorder strength. Our finite-size scaling analysis of entanglement entropy identifies a critical interaction exponent αc≈2α_c \approx 2αc​≈2, which separates a fragile regime, characterized by an exponentially diverging critical disorder, from a robust short-range regime. To rigorously test the system's resistance to avalanches, we couple the boundary to an infinite-temperature bath and track the propagation of the thermalization front into the localized bulk. We find that the characteristic thermalization time follows a unified scaling law, Trth∼exp⁡[κ(α)LW]T_{r_{\text{th}}} \sim \exp[κ(α) LW]Trth​​∼exp[κ(α)LW] (herein, LLL is the system size, and WWW is the disorder intensity), which diverges exponentially with the product of system size and disorder strength. This suppression enables the derivation of a quantitative stability criterion, Wstab(α)W_{\text{stab}}(α)Wstab​(α), representing the minimum critical disorder strength required to maintain avalanche stability. Our results confirm that the MBL phase remains asymptotically stable in the thermodynamic limit when disorder exceeds an interaction-dependent threshold, bridging theoretical debates on long-range MBL and providing a roadmap for observing these dynamics in experimental platforms such as Rydberg atom arrays.
Many-body localization is a profound phase of matter affecting the entire spectrum which emer ges in the presence of disorder in interacting many-body systems. Recently, the stability of many-body localization has been challenged by the avalanche mechanism, in which a small thermal region can spread, destabilizing localization and leading to global thermalization of the system. A key unresolved question is the critical competition between the thermal region's influence and the disorder strength required to trigger such an avalanche. Here, we numerically investigate many-body localization stability in an isolated Heisenberg spin chain of size LLL subjected to a disordered magnetic field. By embedding a tunable thermal region of size PPP, we analyze the system's behavior in both static and dynamical regimes using entanglement entropy and the gap ratio. Our study yields two main findings. Firstly, for strong disorder, the avalanche only occurs if the thermal region scales with system size, specifically when P/LP/LP/L exceeds a threshold value. Secondly, at strong disorder, we identify an intermediate phase between many-body localization and ergodic behavior as PPP increases. This intermediate phase leaves its fingerprint in both static and dynamic properties of the system and tends to vanish in the thermodynamic limit. Although our simulations are restricted to finite system sizes, the analysis suggests that these results hold in the thermodynamic limit for isolated many-body systems.
We investigate the robustness of the many-body localized (MBL) phase to the quantum-avalanche instability by studying the dynamics of a localized spin chain coupled to a $T=\ensuremath{\infty}$ thermal bath through its leftmost site. By analyzing local magnetizations we estimate the size of the thermalized sector of the chain and find that it increases logarithmically slowly in time. This logarithmically slow propagation of the thermalization front allows us to lower-bound the slowest thermalization time, and find a broad parameter range where it scales fast enough with the system size that MBL is robust against thermalization induced by avalanches. The further finding that the imbalance---a global quantity measuring localization---thermalizes over a timescale that is exponential both in disorder strength and system size is in agreement with these results.
Quantum avalanches in Z2\mathbb {Z}_2Z2​-preserving interacting Ising Majorana chain
Recent numerical works have revealed the instability of many-body localized (MBL) phase in disordered quantum many-body systems with finite system sizes and over finite timescales. This instability originates from the rare Griffiths regions in the thermodynamic limit, which rapidly thermalize and induce an avalanche mechanism that destabilizes surrounding MBL regions. Here, we consider the [Formula: see text]-preserving interacting Ising Majorana chain, whose phase diagram is enriched by the presence of an intermediate ergodic phase separating two distinct MBL phases characterized by different long-range orders. We investigate the dynamic characteristics of the model when coupled to an infinite bath under perturbation, and through the scaling behavior of the slowest thermalization rate, we find how the critical disorder strength in finite-size systems is affected by the avalanche mechanism. We also implement the embedded inclusion model and use the time evolution of mutual information between each spin and the artificial Griffiths region to probe the diffusion of the thermal bubble. We observe that in finite-sized systems, the critical disorder strength gradually drifts away from the center. Our results show that both the MBL paramagnetic phase and MBL spin-glass phase are unstable at finite sizes.
We numerically study quantum avalanches in one-dimensional disordered spin systems by attaching two XXZ spin chains. One chain has low disorder representing a rare Griffith's region, or thermal inclusion, and the second has larger disorder, i.e., disorder larger than the observed finite-size crossover. Comparing dynamics of this system to identical systems with uniformly large disorder, we find evidence for exponentially slow thermalization (in disorder) within the many-body localized regime when the rare region is present. We observe a decay of the spin imbalance in the bulk of the large disorder region that persists to long times $(\ensuremath{\sim}{10}^{4})$ and find a universal behavior of the spectral function.
Abstract Avalanches are believed to be the mechanism behind the transition from many-body localization to the thermal phase. We utilize spin chains with constraints to study the physics of quantum avalanches by exact diagonalization of disordered systems coupled to a thermal bath. Single-spin observables are used to characterize localization and quantify the influence of the thermal bath on disordered spin chains. Constraints on the Hilbert space confine the dynamics to a subspace, effectively reducing the Hilbert space dimension and enabling the study of larger systems with limited computational resources. We study the PXP model, and in addition, we construct spin chains with constraints by searching for constraints with a genetic algorithm to reach larger system sizes. We define quantities to measure the strength of avalanches and use these to compare different models. We find that avalanches are more pronounced in models with constraints compared to those without constraints. Results from exact diagonalization are compared with those from studying the Lindblad master equation. We also identify models that exhibit no thermal phases and find stable ergodicity breaking.