Off-diagonal matrix elements of local operators in many-body quantum systems

TL;DR

This paper investigates the statistical properties of off-diagonal matrix elements of local operators in many-body quantum systems, revealing how their distribution changes with integrability and how they scale with system size.

Contribution

It provides a generic analysis of off-diagonal matrix elements in both integrable and non-integrable many-body systems, highlighting the transition in their distribution shape.

Findings

In non-integrable systems, off-diagonal elements follow a Gaussian distribution centered at zero.

Approaching integrability, the distribution becomes a mixture of two Gaussians, sharpening around zero.

The average magnitude of off-diagonal elements scales with system size.

Abstract

In the time evolution of isolated quantum systems out of equilibrium, local observables generally relax to a long-time asymptotic value, governed by the expectation values (diagonal matrix elements) of the corresponding operator in the eigenstates of the system. The temporal fluctuations around this value, response to further perturbations, and the relaxation toward this asymptotic value, are all determined by the off-diagonal matrix elements. Motivated by this non-equilibrium role, we present generic statistical properties of off-diagonal matrix elements of local observables in two families of interacting many-body systems with local interactions. Since integrability (or lack thereof) is an important ingredient in the relaxation process, we analyze models that can be continuously tuned to integrability. We show that, for generic non-integrable systems, the distribution of off-diagonal matrix elements is a gaussian centered at zero. As one approaches integrability, the peak around zero becomes sharper, so that the distribution is approximately a combination of two gaussians. We characterize the proximity to integrability through the deviation of this distribution from a gaussian shape. We also determine the scaling dependence on system size of the average magnitude of off-diagonal matrix elements.