Large deviations in the many-body localization transition: The case of the random-field XXZ chain

TL;DR

This paper investigates the role of rare resonances in the many-body localization transition of the random-field XXZ chain, revealing three regimes and providing a finite-size phase diagram aligned with recent numerical findings.

Contribution

It introduces a mean-field inspired method to analyze rare resonances in Hilbert space, identifying distinct phases and critical disorder levels in the MBL transition.

Findings

Identification of three regimes: ergodic, intermediate, and MBL.

Finite-size phase diagram matching numerical results.

Demonstration that infinitesimal interactions can destabilize Anderson insulators.

Abstract

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.