# Eigenstate thermalization in the two-dimensional transverse field Ising model: II. Off-diagonal matrix elements of observables

## TL;DR

This paper investigates the off-diagonal matrix elements of observables in the eigenstates of the 2D transverse field Ising model, linking quantum chaos to matrix element structure and confirming the eigenstate thermalization hypothesis.

## Contribution

It provides a detailed analysis of off-diagonal matrix elements in the 2D transverse field Ising model, demonstrating the applicability of random matrix theory and ETH in this context.

## Key findings

- Ratio of variances approaches 2 in the quantum chaotic regime
- Off-diagonal matrix elements depend on eigenstate energy differences
- Results support the eigenstate thermalization hypothesis

## Abstract

We study the matrix elements of few-body observables, focusing on the off-diagonal ones, in the eigenstates of the two-dimensional transverse field Ising model. By resolving all symmetries, we relate the onset of quantum chaos to the structure of the matrix elements. In particular, we show that a general result of the theory of random matrices, namely, the value 2 of the ratio of variances (diagonal to off-diagonal) of the matrix elements of Hermitian operators, occurs in the quantum chaotic regime. Furthermore, we explore the behavior of the off-diagonal matrix elements of observables as a function of the eigenstate energy differences, and show that it is in accordance with the eigenstate thermalization hypothesis ansatz.