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

Complexity of dynamics is at the core of quantum many-body chaos and exhibits a hierarchical feature: higher-order complexity implies more chaotic dynamics. Conventional ergodicity in thermalization processes is a manifestation of the lowest order complexity, which is represented by the eigenstate thermalization hypothesis (ETH) stating that individual energy eigenstates are thermal. Here, we propose a higher-order generalization of the ETH, named the $ k $-ETH ($ k=1,2,\dots $), to quantify higher-order complexity of quantum many-body dynamics at the level of individual energy eigenstates, where the lowest order ETH (1-ETH) is the conventional ETH. As a non-trivial contribution of the higher-order ETH, we show that the $ k $-ETH with $ k\geq 2 $ implies a universal behavior of the $ k $th Renyi entanglement entropy of individual energy eigenstates. In particular, the Page correction of the entanglement entropy originates from the higher-order ETH, while as is well known, the volume law can be accounted for by the 1-ETH. We numerically verify that the 2-ETH approximately holds for a nonintegrable system, but does not hold in the integrable case. To further investigate the information-theoretic feature behind the $ k $-ETH, we introduce a concept named a partial unitary $ k $-design (PU $ k $-design), which is an approximation of the Haar random unitary up to the $ k $th moment, where partial means that only a limited number of observables are accessible. The $ k $-ETH is a special case of a PU $ k $-design for the ensemble of Hamiltonian dynamics with random-time sampling. In addition, we discuss the relationship between the higher-order ETH and information scrambling quantified by out-of-time-ordered correlators. Our framework provides a unified view on thermalization, entanglement entropy, and unitary $ k $-designs, leading to deeper characterization of higher-order quantum complexity.