Statistical properties of the off-diagonal matrix elements of observables in eigenstates of integrable systems

TL;DR

This paper investigates the statistical distribution of off-diagonal matrix elements of observables in eigenstates of integrable quantum systems, revealing they follow generalized Gamma distributions under certain conditions.

Contribution

It demonstrates that off-diagonal matrix elements in integrable systems are well described by generalized Gamma distributions, extending understanding of their statistical properties.

Findings

Off-diagonal elements are dense in the XXZ chain but sparse in noninteracting systems.

Distributions are well modeled by generalized Gamma distributions in various conditions.

Localization disrupts the Gamma distribution description.

Abstract

We study the statistical properties of the off-diagonal matrix elements of observables in the energy eigenstates of integrable quantum systems. They have been found to be dense in the spin-1/2 XXZ chain, while they are sparse in noninteracting systems. We focus on the quasimomentum occupation of hard-core bosons in one dimension, and show that the distributions of the off-diagonal matrix elements are well described by generalized Gamma distributions, in both the presence and absence of translational invariance but not in the presence of localization. We also show that the results obtained for the off-diagonal matrix elements of observables in the spin-1/2 XXZ model are well described by a generalized Gamma distribution.