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Physical review. B./Physical review. B

Large-deviation analysis of rare resonances for the many-body localization transition

Giulio Biroli, Alexander K. Hartmann, Marco Tarzia

Published Jul 17, 2024• 22 citations
Publisher

Abstract

The authors explore here the many-body localization (MBL) transition through a novel approach based on an analogy with directed polymers in random media. This study reveals how rare long-range resonances, typically absent in small samples, destabilize the MBL phase in the thermodynamic limit in a broad region of the phase diagram, and shift the transition to much higher disorder levels, offering new insights into quantum avalanches and anomalous transport and relaxation near the transition.

DOI
10.1103/physrevb.110.014205

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The authors explore here the many-body localization (MBL) transition through a novel approach based on an analogy with directed polymers in random media. This study reveals how rare long-range resonances, typically absent in small samples, destabilize the MBL phase in the thermodynamic limit in a broad region of the phase diagram, and shift the transition to much higher disorder levels, offering new insights into quantum avalanches and anomalous transport and relaxation near the transition.
A central theoretical issue at the core of the current research on many-body localization (MBL) consists in characterizing the statistics of rare long-range resonances in many-body eigenstates. This is of paramount importance to understand: (i) the critical properties of the MBL transition and the mechanism for its destabilization through quantum avalanches; (ii) the unusual transport and anomalously slow out-of-equilibrium relaxation when the transition is approached from the metallic side. In order to study and characterize such long-range rare resonances, we develop a large-deviations approach based on an analogy with the physics of directed polymers in random media, and in particular with their freezing glass transition on infinite-dimensional graphs. The basic idea is to enlarge the parameter space by adding an auxiliary parameter (which plays the role of the inverse temperature in the directed polymer formulation) which allows us to fine-tune the effect of anomalously large outliers in the far-tails of the probability distributions of the transmission amplitudes between far-away many-body configurations in the Hilbert space. We first benchmark our approach onto two non-interacting paradigmatic toy models, namely the single-particle Anderson model on the (loop-less) Cayley tree and the Rosenzweig-Porter random matrix ensemble, and then apply it to the study of a class of disordered quantum spin chains in a transverse field. This analysis shows the existence of a broad disorder range in which rare, long-distance resonances, that may form only for a few specific realizations of the disorder and a few specific choice of the random initial state, destabilize the MBL phase, while the genuine MBL transition is shifted to much larger values of the disorder than originally thought.
The effect of rare system-wide resonances in the many-body localization (MBL) transition has recently attracted significant attention. They are expected to play a prominent role in the stability of the MBL phase, prompting the development of new theoretical frameworks to properly account for their statistical weight. We employ a method based on an analogy with mean-field disordered glassy systems to characterize the statistics of transmission amplitudes between distant many-body configurations in Hilbert space, and apply it to the random-field XXZ spin chain. By introducing a Lagrange multiplier, which formally plays the role of an effective temperature controlling the influence of extreme outliers in the heavy-tailed distribution of propagators, we identify three distinct regimes: (i) an ergodic phase with uniform spreading in Hilbert space, (ii) an intermediate regime where delocalization is driven by rare, disorder-dependent long-range resonances, and (iii) a robust MBL phase where such resonances cannot destabilize localization. We derive a finite-size phase diagram in the disorder--interaction plane both in the spin and in the Anderson basis that quantitatively agrees with recent numerical results based on real-space spin-spin correlation functions. We further demonstrate that even infinitesimal interactions can destroy the Anderson insulator at finite disorder, with the critical disorder remaining finite down to small interaction strengths. By visualizing resonant transmission pathways on the Hilbert space graph, we provide a complementary perspective to real-space and spectral probes, revealing how the destabilization of the MBL phase at finite sizes stems from the emergence of resonant paths that become progressively rarer and shorter-ranged deep in the localized phase.
Many-body localization (MBL) is a mechanism by which quantum chains with strong enough disorder can avoid thermalization. Here, the authors define and locate several distinct landmarks having to do with thermalization, rare many-body resonances, and so-called thermal avalanches in systems that can exhibit MBL. These landmarks are well separated in the strength of the disorder, and the estimate of where MBL breaks down in asymptotically large systems is far beyond what is usually pointed to as the MBL transition.
Closed, interacting, quantum systems have the potential to transition to a many-body localized (MBL) phase under the presence of sufficiently strong disorder, hence breaking ergodicity and failing to thermalize. In this work we study the distribution of correlations throughout the ergodic-MBL phase diagram. We find the typical correlations in the MBL phase decay as a stretched exponential with range rrr eventually crossing over to an exponential decay deep in the MBL phase. At the transition, the stretched exponential goes as e−Are^{-A\sqrt{r}}e−Ar​, a decay that is reminiscent of the random singlet phase. While the standard deviation of the log⁡(QMI)\log(QMI)log(QMI) has a range dependence, the log⁡(QMI)\log(QMI)log(QMI) converges to a range-invariant distribution on all other moments (i.e., the skewness and higher) at the transition. The universal nature of these distributions provides distinct phenomenology of the transition different from both the ergodic and MBL phenomenologies. In addition to the typical correlations, we study the extreme correlations in the system, finding that the probability of strong long-range correlations is maximal at the transition, suggesting the proliferation of resonances there. Finally, we analyze the probability that a single bit of information is shared across two halves of a system, finding that this probability is non-zero deep in the MBL phase but vanishes at moderate disorder well above the transition.
The low‐frequency response of systems near the many‐body localization phase transition, on either side of the transition, is dominated by contributions from rare regions that are locally “in the other phase”, i.e., rare localized regions in a system that is typically thermal, or rare thermal regions in a system that is typically localized. Rare localized regions affect the properties of the thermal phase, especially in one dimension, by acting as bottlenecks for transport and the growth of entanglement, whereas rare thermal regions in the localized phase act as local “baths” and dominate the low‐frequency response of the MBL phase. We review recent progress in understanding these rare‐region effects, and discuss some of the open questions associated with them: in particular, whether and in what circumstances a single rare thermal region can destabilize the many‐body localized phase. image
The many-body localization (MBL) problem continues to reveal its rich physical content. Herey, the authors investigate long-range correlations in the random-field Heisenberg chain and find rare, system-spanning ``cat states'' hidden in the high-energy many-body spectrum. These states are identified as nearly degenerate resonant pairs, well captured by an analytical ansatz. Remarkably, such atypical eigenstates persist even at very strong disorder, a regime in which conventional diagnostics suggest robust localization. These results offer fresh insight into the MBL problem and reopen fundamental questions about the mechanisms that might destabilize the MBL phase.