Characterizing complexity of many-body quantum dynamics by higher-order eigenstate thermalization

Characterizing quantum many-body chaos has attracted renewed attention in condensed matter physics, quantum information, and high-energy physics. There are two fundamental concepts regarding this problem. One is the eigenstate thermalization hypothesis (ETH) stating that individual energy eigenstates are thermal. The other is information scrambling, which is quantified by out-of-time-ordered correlators (OTOCs). We propose a higher-order generalization of the ETH, named by the k-ETH (k=1, 2,…) , which provides a unified view on the above two concepts. The lowest order ETH (1-ETH) is the conventional ETH, and the second order ETH (2-ETH) is a sufficient condition of the decay of OTOCs at late times. Our basic idea is that chaotic dynamics share common properties with random unitary dynamics even at a higher level than conventional ergodicity. The k-ETH also implies a universal behavior of the kth-Renyi entanglement entropy of individual energy eigenstates. In particular, we show that the Page correction originates from the higher-order ETH. We numerically verified that the 2-ETH approximately holds for nonintegrable systems, but does not hold for integrable systems.