Many-body localization edge in the random-field Heisenberg chain

TL;DR

This study uses large-scale exact diagonalization to investigate the many-body localization transition in a disordered Heisenberg chain, revealing an energy-dependent mobility edge and characterizing phases by spectral statistics and entanglement.

Contribution

It provides the first energy-resolved analysis of the many-body localization transition, identifying a mobility edge and critical properties in a large system size.

Findings

Existence of a many-body mobility edge separating localized and ergodic phases.

Localized phase exhibits Poisson statistics and area-law entanglement.

Ergodic phase shows Gaussian orthogonal ensemble statistics and volume-law entanglement.

Abstract

We present a large scale exact diagonalization study of the one dimensional spin $1/2$ Heisenberg model in a random magnetic field. In order to access properties at varying energy densities across the entire spectrum for system sizes up to $L=22$ spins, we use a spectral transformation which can be applied in a massively parallel fashion. Our results allow for an energy-resolved interpretation of the many body localization transition including the existence of an extensive many-body mobility edge. The ergodic phase is well characterized by Gaussian orthogonal ensemble statistics, volume-law entanglement, and a full delocalization in the Hilbert space. Conversely, the localized regime displays Poisson statistics, area-law entanglement and non ergodicity in the Hilbert space where a true localization never occurs. We perform finite size scaling to extract the critical edge and exponent of the localization length divergence.