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

TLDR

We conduct a large‑scale exact diagonalization study of the one‑dimensional spin‑½ Heisenberg model in a random magnetic field. We employ a massively parallel spectral transformation to access energy‑resolved properties up to L = 22 spins and perform finite‑size scaling to determine the critical mobility‑edge and localization‑length exponent. Our results reveal an extensive many‑body mobility edge, with the ergodic phase exhibiting GOE statistics, volume‑law entanglement, and full delocalization, while the localized regime shows Poisson statistics, area‑law entanglement, and non‑ergodicity, indicating that true localization never occurs.

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.