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arXiv:1906.10808 · Physical Review Letters

Bounds on Chaos from the Eigenstate Thermalization Hypothesis

Chaitanya Murthy, Mark Srednicki

Published Dec 3, 2019• 135 citations
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Abstract

We show that the known bound on the growth rate of the out-of-time-order four-point correlator in chaotic many-body quantum systems follows directly from the general structure of operator matrix elements in systems that obey the eigenstate thermalization hypothesis. This ties together two key paradigms of thermal behavior in isolated many-body quantum systems.

DOI
10.1103/physrevlett.123.230606

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We show that the known bound on the growth rate of the out-of-time-order four-point correlator in chaotic many-body quantum systems follows directly from the general structure of operator matrix elements in systems that obey the eigenstate thermalization hypothesis. This ties together two key paradigms of thermal behavior in isolated many-body quantum systems.
We consider the statistical properties of eigenstates of the time-evolution operator in chaotic many-body quantum systems. Our focus is on correlations between eigenstates that are specific to spatially extended systems and that characterize entanglement dynamics and operator spreading. In order to isolate these aspects of dynamics from those arising as a result of local conservation laws, we consider Floquet systems in which there are no conserved densities. The correlations associated with scrambling of quantum information lie outside the standard framework established by the eigenstate thermalization hypothesis (ETH). In particular, ETH provides a statistical description of matrix elements of local operators between pairs of eigenstates, whereas the aspects of dynamics we are concerned with arise from correlations among sets of four or more eigenstates. We establish the simplest correlation function that captures these correlations and discuss features of its behavior that are expected to be universal at long distances and low energies. We also propose a maximum-entropy ansatz for the joint distribution of a small number n of eigenstates. In the case n = 2 , this ansatz reproduces ETH. For n = 4 it captures both the growth with time of entanglement between subsystems, as characterized by the purity of the time-evolution operator, and also operator spreading, as characterized by the behavior of the out-of-time-order correlator. We test these ideas by comparing results from Monte Carlo sampling of our ansatz with exact diagonalization studies of Floquet quantum circuits. Published by the American Physical Society 2024
The eigenstate thermalization hypothesis (ETH) implies a form for the matrix elements of local operators between eigenstates of the Hamiltonian, expected to be valid for chaotic systems. Another signal of chaos is a positive Lyapunov exponent, defined on the basis of Loschmidt echo or out of time order correlators. For this exponent to be positive, correlations between matrix elements unrelated by symmetry, usually neglected, have to exist. The same is true for the peak of the dynamic heterogeneity length χ_{4}, relevant for systems with slow dynamics. These correlations, as well as those between elements of different operators, are encompassed in a generalized form of ETH.
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.
The eigenstate thermalization hypothesis explains the emergence of the thermodynamic equilibrium in isolated quantum many-body systems by assuming a particular structure of the observable's matrix elements in the energy eigenbasis. Schematically, it postulates that off-diagonal matrix elements are random numbers and the observables can be described by random matrix theory (RMT). To what extent a RMT description applies, more precisely at which energy scale matrix elements of physical operators become truly uncorrelated, is, however, not fully understood. We study this issue by introducing a novel numerical approach to probe correlations between matrix elements for Hilbert-space dimensions beyond those accessible by exact diagonalization. Our analysis is based on the evaluation of higher moments of operator submatrices, defined within energy windows of varying width. Considering nonintegrable quantum spin chains, we observe that matrix elements remain correlated even for narrow energy windows corresponding to timescales of the order of thermalization time of the respective observables. We also demonstrate that such residual correlations between matrix elements are reflected in the dynamics of out-of-time-ordered correlation functions.
Eigenstate thermalization hypothesis is a detailed statement of the matrix elements of few-body operators in energy eigenbasis of a chaotic Hamiltonian. Part of the statement is that the off-diagonal elements fall exponential for large energy difference. We propose that the exponent (γ>0γ>0γ>0) is a measure of quantum chaos. Smaller γγγ implies more chaotic dynamics. The chaos bound is given by γ=β/4γ=β/4γ=β/4 where βββ is the inverse temperature. We give analytical argument in support of this proposal. The slower exponential fall also means that the action of the operator on a state leads to higher delocalization in energy eigenbasis. Numerically we compare two chaotic Hamiltonians - SYK model and chaotic XXZ spin chain. Using the new measure, we find that the SYK model becomes maximally chaotic at low temperature which has been shown rigorously in previous works. The new measure is more readily accessible compare to other measures using numerical methods.
Out-of-time-order correlators (OTOCs) have become established as a tool to characterise quantum information dynamics and thermalization in interacting quantum many-body systems. It was recently argued that the expected exponential growth of the OTOC is connected to the existence of correlations beyond those encoded in the standard Eigenstate Thermalization Hypothesis (ETH). We show explicitly, by an extensive numerical analysis of the statistics of operator matrix elements in conjunction with a detailed study of OTOC dynamics, that the OTOC is indeed a precise tool to explore the fine details of the ETH. In particular, while short-time dynamics is dominated by correlations, the long-time saturation behavior gives clear indications of an operator-dependent energy scale ω_{GOE} associated to the emergence of an effective Gaussian random matrix theory. We provide an estimation of the finite-size scaling of ω_{GOE} for the general class of observables composed of sums of local operators in the infinite-temperature regime and found linear behavior for the models considered.