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 off-diagonal function in the eigenstate thermalization hypothesis, which is for off-diagonal elements 〈E_{i}|O|E_{j}〉 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 ℏ, a phase-space gradient of the classical Hamiltonian 〈|∇H_{cl}|〉 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 ρ_{dos}^{-1/2} scaling of the off-diagonal function. The analytical predictions are checked numerically in the Lipkin-Meshkov-Glick model.