Signatures of a critical point in the many-body localization transition

TL;DR

This paper identifies a finite-size precursor to the critical point in the many-body localization transition of disordered spin chains, using spectral statistics and fluctuations to distinguish between chaotic and integrable phases.

Contribution

It introduces a finite-size scaling indicator based on kurtosis excess that pinpoints the critical disorder strength in the many-body localization transition.

Findings

Maximum kurtosis excess at the critical point scales linearly with system size.

Spectral statistics transition from random matrix behavior to Poissonian as disorder increases.

The critical point coincides with the energy scale where Thouless and Heisenberg energies match.

Abstract

Disordered interacting spin chains that undergo a many-body localization transition are characterized by two limiting behaviors where the dynamics are chaotic and integrable. However, the transition region between them is not fully understood yet. We propose here a possible finite-size precursor of a critical point that shows a typical finite-size scaling and distinguishes between two different dynamical phases. The kurtosis excess of the diagonal fluctuations of the full one-dimensional momentum distribution from its microcanonical average is maximum at this singular point in the paradigmatic disordered $J_1$-$J_2$ model. For system sizes accessible to exact diagonalization, both the position and the size of this maximum scale linearly with the system size. Furthermore, we show that this singular point is found at the same disorder strength at which the Thouless and the Heisenberg energies coincide. Below this point, the spectral statistics follow the universal random matrix behavior up to the Thouless energy. Above it, no traces of chaotic behavior remain, and the spectral statistics are well described by a generalized semi-Poissonian model, eventually leading to the integrable Poissonian behavior. We provide, thus, an integrated scenario for the many-body localization transition, conjecturing that the critical point in the thermodynamic limit, if it exists, should be given by this value of disorder strength.