Finite size scaling bounds on many-body localized phase transitions

TL;DR

This paper derives bounds on the finite-size scaling exponents of eigenstate phase transitions in disordered quantum systems, specifically focusing on the many-body localization transition, challenging recent numerical findings.

Contribution

It introduces Harris-type bounds on scaling exponents for many-body localization transitions, providing theoretical constraints on eigenstate phase transitions in disordered systems.

Findings

Harris bounds on entanglement entropy scaling exponents

Harris bounds on level statistics scaling exponents

Crossover scales beyond which bounds must hold

Abstract

Quantum phase transitions are usually observed in ground states of correlated systems. Remarkably, eigenstate phase transitions can also occur at finite energy density in disordered, isolated quantum systems. Such transitions fall outside the framework of statistical mechanics as they involve the breakdown of ergodicity. Here, we consider what general constraints can be imposed on the nature of eigenstate transitions due to the presence of disorder. We derive Harris-type bounds on the finite-size scaling exponents of the mean entanglement entropy and level statistics at the many-body localization phase transition using several different arguments. Our results are at odds with recent small-size numerics, for which we estimate the crossover scales beyond which the Harris bound must hold.