Large Deviation Analysis of Eigenstate Thermalization Hypothesis

A plausible mechanism of thermalization in isolated quantum systems is the strong version of the eigenstate thermalization hypothesis (ETH), which states that all the energy eigenstates have thermal properties. In the present work, we propose a new systematic numerical method to test the ETH by focusing on the large deviation property; By using exact numerical diagonalization, we directly calculate the ratio of athermal energy eigenstates in the energy shell. That ratio is exactly zero if the strong ETH is true, while a mathematical theory has only proved that the ratio is at most exponentially small in the system size. Our numerical results confirmed that the strong ETH indeed holds only for non-integrable systems, where we found that the finite-size scaling of the large deviation is double exponential. Furthermore, we numerically verified an expectation that the strong ETH is true for near-integrable systems, even with an infinitely small integrability-breaking term.