Semiclassical analytical solutions of the eigenstate thermalization hypothesis in a quantum billiard

TL;DR

This work extends the eigenstate thermalization hypothesis (ETH) to a single-particle chaotic system by developing a semiclassical framework for the quarter-stadium quantum billiard. The authors derive explicit analytical expressions for the off-diagonal and diagonal ETH components of the observable $\hat{q}_x$ using Weyl symbols and Wigner functions, revealing a local-term plus phase-space-correlation structure and an asymptotic closed form for the $f$ function that predicts bandwidth scaling with energy. They demonstrate that the off-diagonal results are equivalent to Berry's conjecture in the energetic long-wavelength limit, and provide numerical evidence showing agreement with the theory while highlighting fluctuations due to neglected eigenstate correlations. The findings imply ETH-like thermalization features can arise in few-body systems, linking the observed matrix-element structure to underlying classical dynamics and suggesting a concrete interpretation of thermalization as information loss about initial conditions.

Abstract

We derive semiclassical analytical solutions for both the diagonal and off-diagonal functions in the eigenstate thermalization hypothesis (ETH) in a quarter-stadium quantum billiard. For a representative observable, we obtain an explicit expression and an asymptotic closed-form solution that naturally separate into a local contribution and a phase-space correlation term. These analytical results predict the band structure of the observable matrix, including its bandwidth and scaling behavior. We further demonstrate that our analytical formula is equivalent to the prediction of Berry's conjecture. Supported by numerical evidence, we show that Berry's conjecture captures the energetic long-wavelength behavior in the space of eigenstates, while our analytical solution describes the asymptotic behavior of the f function in the semiclassical limit. Finally, by revealing the connection between the bandwidth scaling and the underlying classical dynamics, our results suggest that the ETH carries important physical implications in single-particle and few-body systems, where "thermalization" manifests as the loss of information about initial conditions.

Figures (6)

• Figure 1: Geometry of the quarter-stadium billiard with width $l=2$ and height $h=1$.
• Figure 2: Typical behavior of Eq. \ref{['qx_sq_analy']} as a function of $\Delta E$.
• Figure 3: The red line represents our analytical result from Eq. \ref{['qx_sq']}, while the blue dots show the numerical results for the squared off-diagonal $f$ function.
• Figure 4: The blue dots are the numerical results for diagonal elements $\bra{E_i}\hat{q}_x\ket{E_i}$
• Figure 5: Here is the result for $\langle A_i(\bm{p}_1)A_j(\bm{p}_2)\rangle$ while setting $i=j$. They are plotted with respect to $|\bm{p}_1-\bm{p}_2|$, for labeling the distance between plane wave bases. The average is taken over an energy shell consisting $N_{ave}$ energy levels.