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Eigenstate thermalization hypothesis and eigenstate-to-eigenstate fluctuations

We investigate the extent to which the eigenstate thermalization hypothesis~(ETH) is valid or violated in the non-integrable and the integrable spin-$1/2$ XXZ chain. We perform the energy-resolved analysis of the statistical properties of matrix elements $\{O_{γα}\}$ of an observable $\hat{O}$ in the energy eigenstate basis. The Hilbert space is coarse-grained into energy shells of width $Δ_E$, with which one can define a block submatrix $\tilde{O}^{(b,a)}$ consisting of elements between eigenstates in the $a$th and $b$th shells. Each block submatrix is characterized by constant values of $E_{γα}=(E_γ+E_α)/2 \simeq \tilde{E}$ and $ω_{γα}= (E_γ-E_α) \simeq ω$ up to $Δ_E$. We will show that all matrix elements within a block are statistically equivalent to each other in the non-integrable case. Their distribution is characterized by $\bar{E}$ and $ω$, and follows the prediction of the ETH. In stark contrast, eigenstate-to-eigenstate fluctuations persist in the integrable case. Consequently, matrix elements $O_{γα}$ cannot be characterized by the energy parameters $E_{γα}$ and $ω_{γα}$ 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.