An operator-Weyl-symbol approach to eigenstate thermalization hypothesis

In this letter, by an approach that employs Weyl symbols for operators, a semiclassical theory is developed for the offdiagonal function in the eigenstate thermalization hypothesis, which is for offdiagonal elements $\langle{E_i}\left|O\right|{E_j}\rangle$ of an observable $O$ on the energy basis. It is shown analytically that the matrix of $O$ has a banded structure, possessing a bandwidth $w_b$ that scales linearly with $\hbar$, a phase-space gradient of the classical Hamiltonian, $\langle\left|{\boldsymbol{\nabla }H_{\rm cl}}\right|\rangle$, and an $O$-dependent property. This predicts that the thermalization timescale of a quantum system may be inversely proportional to the phase-space gradient of the Hamiltonian, aligning with intuitions in classical thermalization. This approach also elucidates the origin of a $\rho_{\rm dos}^{-1/2}$-scaling of the offdiagonal function. The analytical predictions are checked numerically in the Lipkin-Meshkov-Glick model.