Eigenstate thermalization in dual-unitary quantum circuits: Asymptotics of spectral functions

TL;DR

This paper derives an exact asymptotic expression for spectral functions in dual-unitary quantum circuits, confirming Gaussian distribution of matrix elements and providing insights into thermalization mechanisms in these systems.

Contribution

It introduces a precise asymptotic formula for spectral functions in dual-unitary circuits, linking matrix element distributions to dynamical correlations and thermalization.

Findings

Excellent agreement between asymptotic formula and exact diagonalization results.

Confirmed Gaussian distribution of matrix elements through numerical higher moments.

Finite size fluctuations are connected to intermediate-time dynamical correlations.

Abstract

The eigenstate thermalization hypothesis provides to date the most successful description of thermalization in isolated quantum systems by conjecturing statistical properties of matrix elements of typical operators in the (quasi-)energy eigenbasis. Here we study the distribution of matrix elements for a class of operators in dual-unitary quantum circuits in dependence of the frequency associated with the corresponding eigenstates. We provide an exact asymptotic expression for the spectral function, i.e., the second moment of this frequency resolved distribution. The latter is obtained from the decay of dynamical correlations between local operators which can be computed exactly from the elementary building blocks of the dual-unitary circuits. Comparing the asymptotic expression with results obtained by exact diagonalization we find excellent agreement. Small fluctuations at finite system size are explicitly related to dynamical correlations at intermediate times and the deviations from their asymptotical dynamics. Moreover, we confirm the expected Gaussian distribution of the matrix elements by computing higher moments numerically.