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

Area of Science:

• Quantum mechanics
• Statistical mechanics
• Condensed matter physics

Background:

• The eigenstate thermalization hypothesis (ETH) is crucial for understanding thermalization in many-body quantum systems.
• ETH relies on two key functions: one for averaged diagonal elements and another for the variance of off-diagonal elements of an observable.
• Existing semiclassical expressions for ETH are limited, particularly for the off-diagonal elements.

Purpose of the Study:

• To derive higher-order semiclassical expressions for the diagonal and off-diagonal functions relevant to ETH.
• To provide analytical insights into the behavior of these functions beyond the zeroth order of ℏ.
• To numerically validate the derived expressions using a specific physical model.

Main Methods:

• Development of semiclassical approximations for ETH-related functions.
• Derivation of higher-order corrections in ℏ for the diagonal elements.
• Analytical derivation for the scale of the central platform height of the off-diagonal elements.
• Numerical verification in the Lipkin-Meshkov-Glick model.

Main Results:

• A semiclassical expression for the averaged diagonal elements, including higher-order ℏ contributions, was derived.
• A novel semiclassical expression for the off-diagonal elements' central platform height scale was obtained.
• The analytical results were successfully validated through numerical simulations in the Lipkin-Meshkov-Glick model.

Conclusions:

• The study provides enhanced analytical tools for investigating thermalization via ETH.
• The derived semiclassical expressions offer deeper insights into the behavior of diagonal and off-diagonal elements.
• The findings pave the way for future theoretical and numerical investigations of thermalization processes.