Spectral statistics across the many-body localization transition

TL;DR

This paper analyzes the evolution of spectral statistics across the many-body localization transition, revealing a two-stage process from Wigner-Dyson to semi-Poisson statistics through fractal matrix elements and plasma models.

Contribution

It introduces a two-stage framework for spectral statistics transition across the MBLT, linking matrix element fractality to level statistics universality classes.

Findings

Flow from Wigner-Dyson to Poisson is two-stage

Fractal matrix elements induce power-law interactions

Level statistics follow a semi-Poisson universality class

Abstract

The many-body localization transition (MBLT) between ergodic and many-body localized phase in disordered interacting systems is a subject of much recent interest. Statistics of eigenenergies is known to be a powerful probe of crossovers between ergodic and integrable systems in simpler examples of quantum chaos. We consider the evolution of the spectral statistics across the MBLT, starting with mapping to a Brownian motion process that analytically relates the spectral properties to the statistics of matrix elements. We demonstrate that the flow from Wigner-Dyson to Poisson statistics is a two-stage process. First, fractal enhancement of matrix elements upon approaching the MBLT from the metallic side produces an effective power-law interaction between energy levels, and leads to a plasma model for level statistics. At the second stage, the gas of eigenvalues has local interaction and level statistics belongs to a semi-Poisson universality class. We verify our findings numerically on the XXZ spin chain. We provide a microscopic understanding of the level statistics across the MBLT and discuss implications for the transition that are strong constraints on possible theories.