Many-body localization and thermalization in the full probability distribution function of observables

TL;DR

This paper explores how many-body localization affects thermalization in quantum systems by analyzing the full distribution functions of observables after a quench, revealing that complete integrability breaking leads to equilibrium-like distributions.

Contribution

It demonstrates that the full probability distribution functions of observables reflect the transition from localization to delocalization in quasiparticle space after integrability breaking.

Findings

Distribution functions qualitatively match equilibrium at long times.

Quantitative agreement occurs only when integrability is fully broken.

Eigenstates become diffusive in quasiparticle space with full integrability breaking.

Abstract

We investigate the relation between thermalization following a quantum quench and many-body localization in quasiparticle space in terms of the long-time full distribution function of physical observables. In particular, expanding on our recent work [E. Canovi {\em et al.}, Phys. Rev. B {\bf 83}, 094431 (2011)], we focus on the long-time behavior of an integrable XXZ chain subject to an integrability-breaking perturbation. After a characterization of the breaking of integrability and the associated localization/delocalization transition using the level spacing statistics and the properties of the eigenstates, we study the effect of integrability-breaking on the asymptotic state after a quantum quench of the anisotropy parameter, looking at the behavior of the full probability distribution of the transverse and longitudinal magnetization of a subsystem. We compare the resulting distributions with those obtained in equilibrium at an effective temperature set by the initial energy. We find that, while the long time distribution functions appear to always agree {\it qualitatively} with the equilibrium ones, {\it quantitative} agreement is obtained only when integrability is fully broken and the relevant eigenstates are diffusive in quasi-particle space.