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 ($k=1,2,\cdots$), 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. The explicit condition of the $k$-ETH is obtained by comparing Hamiltonian dynamics with the Haar random unitary of the $k$-fold channel. As a nontrivial contribution of the higher-order ETH, we show that the $k$-ETH with $k\ge 2$ implies a universal behavior of the $k\mathrm{th}$ Rényi 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 $k\mathrm{th}$ 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.

• Received 4 December 2019
• Accepted 24 March 2020

DOI:https://doi.org/10.1103/PhysRevA.101.042126