A constructive theory of the numerically accessible many-body localized to thermal crossover

TL;DR

This paper presents a model explaining the MBL to thermal crossover in short 1D chains, reproducing key numerical observations and suggesting the numerics are consistent with a true MBL phase in the thermodynamic limit.

Contribution

The authors develop a constructive model capturing the MBL-thermal crossover, aligning with numerical data and clarifying the nature of the transition.

Findings

Model reproduces correlation length exponent $ u=1$

Explains exponential growth of Thouless time with disorder

Resonance behavior matches numerical observations

Abstract

The many-body localised (MBL) to thermal crossover observed in exact diagonalisation studies remains poorly understood as the accessible system sizes are too small to be in an asymptotic scaling regime. We develop a model of the crossover in short 1D chains in which the MBL phase is destabilised by the formation of many-body resonances. The model reproduces several properties of the numerically observed crossover, including an apparent correlation length exponent $\nu=1$, exponential growth of the Thouless time with disorder strength, linear drift of the critical disorder strength with system size, scale-free resonances, apparent $1/\omega$ dependence of disorder-averaged spectral functions, and sub-thermal entanglement entropy of small subsystems. In the crossover, resonances induced by a local perturbation are rare at numerically accessible system sizes $L$ which are smaller than a \emph{resonance length} $\lambda$. For $L \gg \sqrt{\lambda}$, resonances typically overlap, and this model does not describe the asymptotic transition. The model further reproduces controversial numerical observations which Refs. [\v{S}untajs et al, 2019] and [Sels & Polkovnikov, 2020] claimed to be inconsistent with MBL. We thus argue that the numerics to date is consistent with a MBL phase in the thermodynamic limit.