Exact infinite-time statistics of the Loschmidt echo for a quantum quench

TL;DR

This paper derives an exact long-time distribution for the Loschmidt echo in a quantum XY chain after a quench, revealing Gaussian behavior in the thermodynamic limit and universal double-peaked distributions in the quasi-critical regime.

Contribution

It provides an exact analytical expression for the Loschmidt echo distribution at long times, extending results to general models in the small-quench regime.

Findings

Logarithm of Loschmidt echo is normally distributed in the thermodynamic limit.

Distribution becomes double-peaked in the quasi-critical small-quench regime.

Results are obtained via a central limit theorem-type approach.

Abstract

The equilibration dynamics of a closed quantum system is encoded in the long-time distribution function of generic observables. In this paper we consider the Loschmidt echo generalized to finite temperature, and show that we can obtain an exact expression for its long-time distribution for a closed system described by a quantum XY chain following a sudden quench. In the thermodynamic limit the logarithm of the Loschmidt echo becomes normally distributed, whereas for small quenches in the opposite, quasi-critical regime, the distribution function acquires a universal double-peaked form indicating poor equilibration. These findings, obtained by a central limit theorem-type result, extend to completely general models in the small-quench regime.