Low-frequency behavior of off-diagonal matrix elements in the integrable XXZ chain and in a locally perturbed quantum-chaotic XXZ chain

TL;DR

This paper investigates the low-frequency behavior of off-diagonal matrix elements in the integrable XXZ chain and a perturbed quantum-chaotic XXZ chain, revealing distinct behaviors related to integrability and chaos.

Contribution

It provides a detailed comparison of the low-frequency variances of off-diagonal matrix elements in integrable and quantum-chaotic XXZ models, highlighting differences in their dynamical properties.

Findings

In the integrable model, variances vanish or remain nonzero as frequency approaches zero.

In the quantum-chaotic model, variances remain nonzero at low frequencies, indicating diffusive dynamics.

Distinct properties of local operator matrix elements differentiate integrable from chaotic systems.

Abstract

We study the matrix elements of local operators in the eigenstates of the integrable XXZ chain and of the quantum-chaotic model obtained by locally perturbing the XXZ chain with a magnetic impurity. We show that, at frequencies that are polynomially small in the system size, the behavior of the variances of the off-diagonal matrix elements can be starkly different depending on the operator. In the integrable model we find that, as the frequency $\omega\rightarrow0$, the variances are either nonvanishing (generic behavior) or vanishing (for a special class of operators). In the quantum-chaotic model, on the other hand, we find the variances to be nonvanishing as $\omega\rightarrow0$ and to indicate diffusive dynamics. We highlight which properties of the matrix elements of local operators are different between the integrable and quantum-chaotic models independently of the specific operator selected.