Eigenstate thermalization hypothesis and eigenstate-to-eigenstate fluctuations

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