Quantifying quantum ergodicity with semiclassical ℏ-expansion in Wigner-Weyl quantization

The ℏ-expansion of the Wigner function provides a pathway to evaluate quantum corrections of observables with semiclassical methods. In this work, we apply this method to different integrable and chaotic systems in a Gaussian ensemble with an energy width of a few mean level spacings. Examples of the systems we present include quartic and cosine potentials in 1D, 2D harmonic oscillators with nonlinear couplings and with a random Lorentz gas scattering potential, and the Fermi–Pasta–Ulam–Tsingou (FPUT) model. For 1D systems, quantum corrections of O(ℏ²) improve the error in energy levels and observables by several orders of magnitudes, and the energy window width can be down to a few level spacings. For chaotic but non-ergodic systems, the level-to-level quantum fluctuation is large, and the first few orders in corrections only work at larger energy widths. For many-particle systems, the Wigner function method gives us information about local observables and their correlation functions which would be inaccessible via exact diagonalization. By studying the observables and how corrections of them in different orders depend on the energy width, we can obtain a coarse-graining picture of the level clustering behavior in chaotic systems, thus quantitatively evaluating their degrees of ergodicity.