Single-particle eigenstate thermalization in quantum-chaotic quadratic Hamiltonians

TL;DR

This paper investigates eigenstate thermalization in single-particle eigenstates of quantum-chaotic quadratic Hamiltonians, demonstrating thermalization properties and statistical behaviors of matrix elements in models like SYK2 and Anderson.

Contribution

It provides a detailed analysis of eigenstate thermalization in quadratic models, revealing the statistical properties of matrix elements and identifying conditions for Gaussian distributions.

Findings

Diagonal matrix elements have fluctuations vanishing with system size.

Variance ratio of diagonal to off-diagonal elements is 2, matching random matrix theory.

Distributions of matrix elements are not always Gaussian, but some observables exhibit Gaussian behavior.

Abstract

We study the matrix elements of local and nonlocal operators in the single-particle eigenstates of two paradigmatic quantum-chaotic quadratic Hamiltonians; the quadratic Sachdev-Ye-Kitaev (SYK2) model and the three-dimensional Anderson model below the localization transition. We show that they display eigenstate thermalization for normalized observables. Specifically, we show that the diagonal matrix elements exhibit vanishing eigenstate-to-eigenstate fluctuations, and a variance proportional to the inverse Hilbert space dimension. We also demonstrate that the ratio between the variance of the diagonal and the off-diagonal matrix elements is $2$, as predicted by the random matrix theory. We study distributions of matrix elements of observables and establish that they need not be Gaussian. We identify the class of observables for which the distributions are Gaussian.