Power-Law Random Banded Matrix Ensemble as the Effective Model for Many-Body Localization Transition

TL;DR

This paper demonstrates that the power-law random band matrix ensemble effectively models the many-body localization transition, accurately reproducing spectral statistics and eigenvector properties with less computational effort.

Contribution

The study introduces the PRBM ensemble as a simplified, effective model for MBL transition, matching key spectral and eigenvector features of physical systems.

Findings

PRBM reproduces eigenvalue statistics across the phase diagram

Eigenvector entanglement entropy transitions from volume-law to area-law

Critical exponent for transition is approximately 0.83

Abstract

We employ the power-law random band matrix (PRBM) ensemble with single tuning parameter $\mu $ as the effective model for many-body localization (MBL) transition in random spin systems. We show the PRBM accurately reproduce the eigenvalue statistics on the entire phase diagram through the fittings of high-order spacing ratio distributions $P(r^{(n)})$ as well as number variance $\Sigma ^{2}(l)$, in systems both with and without time-reversal symmetry. For the properties of eigenvectors, it's shown the entanglement entropy of PRBM displays an evolution from volume-law to area-law behavior which signatures an ergodic-MBL transition, and the critical exponent is found to be $\nu =0.83\pm 0.15$, close to the value obtained in 1D physical model by exact diagonalization while the computational cost here is much less.