Semiclassical study of diagonal and off-diagonal functions in the eigenstate thermalization hypothesis

The so-called eigenstate thermalization hypothesis (ETH), which has been tested in various many-body models by numerical simulations, supplies a way of understanding eventual thermalization and is believed to be important for understanding processes of thermalization. Two functions play important roles in the application of ETH, one for averaged diagonal elements and the other for the variance of off-diagonal elements of an observable addressed by ETH on the energy basis. For the former function, a semiclassical expression is known of the zeroth order of ℏ, while, little is known analytically for the latter. In this paper, a semiclassical expression is derived for the former diagonal function, which includes higher-order contributions of ℏ. And, another semiclassical expression is derived for the scale of the height of a central platform of the latter off-diagonal function. These analytical results are checked numerically in the Lipkin-Meshkov-Glick model. Possible relevance of the latter result for future investigation is discussed.