Resonance Proliferation Across Localization Transitions

TL;DR

This paper develops a statistical method to predict how many-body resonances proliferate across localization transitions, explaining finite-size drifts in many-body localization models.

Contribution

It introduces a flow equation for resonance density that captures the transition from localized to delocalized phases, validated on multiple models.

Findings

The flow equation accurately predicts resonance proliferation in models.

The predicted resonance distribution matches numerical results.

Connects resonance flow to eigenstate properties and dynamics.

Abstract

Models of many-body localization (MBL) exhibit slow numerical drifts towards delocalization with increasing system size, for which no satisfactory theory exists. Numerics indicates that these drifts are driven by the proliferation of many-body resonances at intermediate disorder strengths. We develop a statistical method to predict the distribution of resonance oscillation frequencies which captures how the formation of resonances at larger frequency scales subsequently affects the formation of resonances at lower frequencies. Working within the statistical Jacobi approximation (SJA), we derive a flow equation for a power-law exponent $\theta(w)$ characterizing the density of resonances at frequency scale $w$. A localized phase is described by a line of fixed points with $\theta(w)>0$, while an instability of the flow signals resonance proliferation and the onset of thermalization. The predicted $\theta(w)$ matches numerics on the Anderson model on random regular graphs and the L\'evy-Rosenzweig-Porter random matrix ensemble, both of which host resonance-driven delocalization transitions. We further connect the flow to eigenstate properties such as the participation ratio and to dynamical observables such as the return probability. The predicted $\theta(w)$ also matches what is numerically measured in real-space models of MBL at intermediate disorder strengths, representing a significant step towards explaining the finite-size drifts observed in MBL.