We investigate the extent to which the eigenstate thermalization hypothesis (ETH) is valid or violated in the nonintegrable and the integrable spin-$1/2 XXZ$ chains. We perform the energy-resolved analysis of statistical properties of matrix elements of observables in the energy eigenstate basis. The Hilbert space is divided into energy shells of constant width, and a block submatrix is constructed whose columns and rows correspond to the eigenstates in the respective energy shells. In each submatrix, we measure the second moment of off-diagonal elements in a column. The columnar second moments are distributed with a finite variance for finite-sized systems. We show that the relative variance of the columnar second moments decreases as the system size increases in the non-integrable system. The self-averaging behavior indicates that the energy eigenstates are statistically equivalent to each other, which is consistent with the ETH. In contrast, the relative variance does not decrease with the system size in the integrable system. The persisting eigenstate-to-eigenstate fluctuation implies that the matrix elements cannot be characterized with the energy parameters only. Our result explains the origin for the breakdown of the fluctuation dissipation theorem in the integrable system. The eigenstate-to-eigenstate fluctuations sheds a new light on the meaning of the ETH.

中文翻译：

---

本征态热化假设和本征态到本征态波动

我们调查了本征态热化假设（ETH）在不可积和可积自旋中的有效程度或违反的程度，$\mathrm{1个}/2 XXž$链。我们以能量本征态为基础，对可观察物的矩阵元素的统计特性进行能量分解分析。希尔伯特空间被划分为等宽的能量壳，并构建了一个块子矩阵，其子行和列对应于各个能量壳中的本征态。在每个子矩阵中，我们测量列中非对角元素的第二矩。对于有限大小的系统，柱状第二矩具有有限的方差分布。我们表明，在不可积系统中，柱状第二矩的相对方差随着系统大小的增加而减小。自平均行为表明能量本征态在统计上彼此相等，这与ETH一致。相反，相对方差不会随着可积系统中系统大小的减小而减小。持续的本征态到本征态波动意味着矩阵元素不能仅用能量参数来表征。我们的结果解释了可积系统中波动耗散定理分解的根源。本征态到本征态的波动为ETH的含义提供了新的视角。