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arXiv:1911.10755 · Physical Review A

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

Kazuya Kaneko, Eiki Iyoda, Takahiro Sagawa

Published Apr 27, 2020• 38 citations
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Abstract

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 ($k=1,2,\ensuremath{\cdots}$), 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 $k\ensuremath{\ge}2$ implies a universal behavior of the $k\mathrm{th}$ R\'enyi 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.

DOI
10.1103/physreva.101.042126

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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 kkk-ETH (k=1,2,\ensuremath⋯k=1,2,\ensuremath{\cdots}k=1,2,\ensuremath⋯), 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 kkk-ETH is obtained by comparing Hamiltonian dynamics with the Haar random unitary of the kkk-fold channel. As a nontrivial contribution of the higher-order ETH, we show that the kkk-ETH with k\ensuremath≥2k\ensuremath{\ge}2k\ensuremath≥2 implies a universal behavior of the kthk\mathrm{th}kth R'enyi 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 kkk-ETH, we introduce a concept named a partial unitary kkk-design (PU kkk-design), which is an approximation of the Haar random unitary up to the kthk\mathrm{th}kth moment, where ``partial'' means that only a limited number of observables are accessible. The kkk-ETH is a special case of a PU kkk-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 kkk-designs, leading to deeper characterization of higher-order quantum complexity.
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The eigenstate thermalization hypothesis (ETH) provides a powerful framework for understanding thermalization in isolated quantum many-body systems, yet a complete and conceptually transparent derivation has remained elusive. In this work, we derive the structure of ETH from a minimal dynamical principle, which we term the eigenstate typicality principle (ETP), together with general kinematic ingredients arising from entropy maximization, Hilbert-space geometry, and locality. ETP asserts that in quantum-chaotic systems, energy eigenstates are statistically indistinguishable, with respect to local measurements, from states drawn from the Haar measure on a narrow microcanonical shell. Within this framework, diagonal ETH arises from concentration of measure, provided that eigenstate typicality holds. The structure of off-diagonal matrix elements is then fixed by entropic scaling and the finite-time dynamical correlations of local observables, with ETP serving as the dynamical bridge to energy eigenstates, without invoking random-matrix assumptions. Our results establish ETH as a consequence of entropy, Hilbert-space geometry, and chaos-induced eigenstate typicality, and clarify its regime of validity across generic quantum-chaotic many-body systems, thereby deepening our understanding of quantum thermalization and the emergence of statistical mechanics from unitary many-body dynamics.
We provide a pedagogical introduction to eigenstate thermalization. This phenomenon, which occurs in generic quantum many-body systems, allows one to understand why thermalization takes place in isolated systems under unitary dynamics. We motivate eigenstate thermalization using random matrix theory and discuss recent complementary results for the volume-law entanglement entropy of Haar-random states. We discuss numerical results that highlight the corresponding behaviors in quantum many-body systems.
Abstract Scrambling is a process by which the state of a quantum system is effectively randomized due to the global entanglement that “hides” initially localized quantum information. Closely related notions include quantum chaos and thermalization. Such phenomena play key roles in the study of quantum gravity, many-body physics, quantum statistical mechanics, quantum information etc. Scrambling can exhibit different complexities depending on the degree of randomness it produces. For example, notice that the complete randomization implies scrambling, but the converse does not hold; in fact, there is a significant complexity gap between them. In this work, we lay the mathematical foundations of studying randomness complexities beyond scrambling by entanglement properties. We do so by analyzing the generalized (in particular Rényi) entanglement entropies of designs, i.e. ensembles of unitary channels or pure states that mimic the uniformly random distribution (given by the Haar measure) up to certain moments. A main collective conclusion is that the Rényi entanglement entropies averaged over designs of the same order are almost maximal. This links the orders of entropy and design, and therefore suggests Rényi entanglement entropies as diagnostics of the randomness complexity of corresponding designs. Such complexities form a hierarchy between information scrambling and Haar randomness. As a strong separation result, we prove the existence of (state) 2-designs such that the Rényi entanglement entropies of higher orders can be bounded away from the maximum. However, we also show that the min entanglement entropy is maximized by designs of order only logarithmic in the dimension of the system. In other words, logarithmic-designs already achieve the complexity of Haar in terms of entanglement, which we also call max-scrambling. This result leads to a generalization of the fast scrambling conjecture, that max-scrambling can be achieved by physical dynamics in time roughly linear in the number of degrees of freedom. This paper is an extended version of Phys. Rev. Lett. 120 (2018) 130502 [1].
The eigenstate thermalization hypothesis (ETH), which asserts that every eigenstate of a many-body quantum system is indistinguishable from a thermal ensemble, plays a pivotal role in understanding thermalization of isolated quantum systems. Yet, no evidence has been obtained as to whether the ETH holds for all few-body operators in a chaotic system; such few-body operators include key quantities in statistical mechanics, such as the total magnetization, the momentum distributions, and their low-order thermal and quantum fluctuations. Here, we formulate a conjecture that for a generic nonintegrable system the ETH holds simultaneously for all mmm-body operators with m<α∗Nm < α_{\ast} Nm<α∗​N in the thermodynamic limit for some nonzero constant α∗>0α_{\ast} > 0α∗​>0. We first show the existence of such nontrivial constants for idealized (pseudo) random-matrix descriptions of many-body eigenstates. We then verify the conjecture for generic spin, Bose, and Fermi systems with local and few-body interactions by large-scale numerical calculations. Our results imply that generic systems satisfy the ETH simultaneously for all few-body operators, including their thermal and quantum fluctuations.
It is now well-established that classical statistical mechanics emerges from the entanglement structure of quantum chaotic systems, as quantified by the (subsystem) eigenstate thermalization hypothesis (ETH). While this statement rests on the well-studied bipartite entanglement of the eigenstates of such systems, recent years have shown that many-body states are often characterized by multipartite entanglement, raising the question: what are the universal multipartite entanglement features of chaotic eigenstates? We answer this question in part by studying a tripartite generalization of the ergodic bipartition ansatz through the lens of two tripartite entanglement probes, the Rényi tripartite multi-entropy and the Rényi reflected entropy. We derive leading-order analytical expressions for the ensemble averages of both quantities using a saddle-point analysis. Beyond these measures, we obtain the average of the Rényi-2 reflected density matrix, ρAA∗(2)ρ_{AA^*}^{(2)}ρAA∗(2)​,which controls the Rényi-2 reflected entropy. We show that ρAA∗(2)ρ_{AA^*}^{(2)}ρAA∗(2)​ in a chaotic eigenstate is well-described by this ensemble average which, unlike an ordinary reduced density matrix, takes three qualitatively distinct forms across subsystem-size regimes, which may be viewed as a tripartite generalization of subsystem ETH. We benchmark our analytical predictions against exact diagonalization of the mixed-field Ising model.