Eigenstate thermalization hypothesis beyond standard indicators: Emergence of random-matrix behavior at small frequencies

TL;DR

This paper investigates the transition from correlated to uncorrelated matrix elements in nonintegrable quantum spin chains, revealing a scale-dependent emergence of random-matrix behavior consistent with the eigenstate thermalization hypothesis.

Contribution

It demonstrates that eigenvalue distributions can detect correlations between matrix elements and identifies a scale where matrix elements become effectively uncorrelated, indicating a transition to random-matrix behavior.

Findings

Eigenvalue distribution at fixed energy density reveals correlations between matrix elements.

At smaller energy scales, eigenvalue distribution approaches a semicircle, indicating random-matrix behavior.

Standard indicators of ETH do not fully capture the correlations present in matrix elements.

Abstract

Using numerical exact diagonalization, we study matrix elements of a local spin operator in the eigenbasis of two different nonintegrable quantum spin chains. Our emphasis is on the question to what extent local operators can be represented as random matrices and, in particular, to what extent matrix elements can be considered as uncorrelated. As a main result, we show that the eigenvalue distribution of band submatrices at a fixed energy density is a sensitive probe of the correlations between matrix elements. We find that, on the scales where the matrix elements are in a good agreement with all standard indicators of the eigenstate thermalization hypothesis, the eigenvalue distribution still exhibits clear signatures of the original operator, implying correlations between matrix elements. Moreover, we demonstrate that at much smaller energy scales, the eigenvalue distribution approximately assumes the universal semicircle shape, indicating transition to the random-matrix behavior, and in particular that matrix elements become uncorrelated.