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 Eᵢ|O|Eⱼ of an observable O on the energy basis. It is shown analytically that the matrix of O has a banded structure, possessing a bandwidth wb that scales linearly with , a phase-space gradient of the classical Hamiltonian, |∇ Hcl|, 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 ρdos-1/2-scaling of the offdiagonal function. The analytical predictions are checked numerically in the Lipkin-Meshkov-Glick model.