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arXiv:2401.13829 · Journal of High Energy Physics

Generalized free cumulants for quantum chaotic systems

Siddharth Jindal, Pavan Hosur

Published Sep 12, 2024• 14 citations

Abstract

A bstract The eigenstate thermalization hypothesis (ETH) is the leading conjecture for the emergence of statistical mechanics in generic isolated quantum systems and is formulated in terms of the matrix elements of operators. An analog known as the ergodic bipartition (EB) describes entanglement and locality and is formulated in terms of the components of eigenstates. In this paper, we significantly generalize the EB and unify it with the ETH, extending the EB to study higher correlations and systems out of equilibrium. Our main result is a diagrammatic formalism that computes arbitrary correlations between eigenstates and operators based on a recently uncovered connection between the ETH and free probability theory. We refer to the connected components of our diagrams as generalized free cumulants. We apply our formalism in several ways. First, we focus on chaotic eigenstates and establish the so-called subsystem ETH and the Page curve as consequences of our construction. We also improve known calculations for thermal reduced density matrices and comment on an inherently free probabilistic aspect of the replica approach to entanglement entropy previously noticed in a calculation for the Page curve of an evaporating black hole. Next, we turn to chaotic quantum dynamics and demonstrate the ETH as a sufficient mechanism for thermalization, in general. In particular, we show that reduced density matrices relax to their equilibrium form and that systems obey the Page curve at late times. We also demonstrate that the different phases of entanglement growth are encoded in higher correlations of the EB. Lastly, we examine the chaotic structure of eigenstates and operators together and reveal previously overlooked correlations between them. Crucially, these correlations encode butterfly velocities, a well-known dynamical property of interacting quantum systems.

DOI: 10.1007/jhep09(2024)066

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A bstract The eigenstate thermalization hypothesis (ETH) is the leading conjecture for the emergence of statistical mechanics in generic isolated quantum systems and is formulated in terms of the matrix elements of operators. An analog known as the ergodic bipartition (EB) describes entanglement and locality and is formulated in terms of the components of eigenstates. In this paper, we significantly generalize the EB and unify it with the ETH, extending the EB to study higher correlations and systems out of equilibrium. Our main result is a diagrammatic formalism that computes arbitrary correlations between eigenstates and operators based on a recently uncovered connection between the ETH and free probability theory. We refer to the connected components of our diagrams as generalized free cumulants. We apply our formalism in several ways. First, we focus on chaotic eigenstates and establish the so-called subsystem ETH and the Page curve as consequences of our construction. We also improve known calculations for thermal reduced density matrices and comment on an inherently free probabilistic aspect of the replica approach to entanglement entropy previously noticed in a calculation for the Page curve of an evaporating black hole. Next, we turn to chaotic quantum dynamics and demonstrate the ETH as a sufficient mechanism for thermalization, in general. In particular, we show that reduced density matrices relax to their equilibrium form and that systems obey the Page curve at late times. We also demonstrate that the different phases of entanglement growth are encoded in higher correlations of the EB. Lastly, we examine the chaotic structure of eigenstates and operators together and reveal previously overlooked correlations between them. Crucially, these correlations encode butterfly velocities, a well-known dynamical property of interacting quantum systems.

Quantum thermalization is well understood via the eigenstate thermalization hypothesis (ETH). The general form of ETH, describing all the relevant correlations of matrix elements, may be derived on the basis of a "typicality" argument of invariance with respect to local rotations involving nearby energy levels. In this Letter, we uncover the close relation between this perspective on ETH and free probability theory, as applied to a thermal ensemble or an energy shell. This mathematical framework allows one to reduce in a straightforward way higher-order correlation functions to a decomposition given by minimal blocks, identified as free cumulants, for which we give an explicit formula. This perspective naturally incorporates the consistency property that local functions of ETH operators also satisfy ETH. The present results uncover a direct connection between the eigenstate thermalization hypothesis and the structure of free probability, widening considerably the latter's scope and highlighting its relevance to quantum thermalization.

The eigenstate thermalization hypothesis (ETH) has been established as the general framework to understand quantum statistical mechanics. Only recently has the attention been paid to so-called full ETH, which accounts for higher-order correlations among matrix elements, and that can be rationalized theoretically using the language of free probability. In this work, we perform the first numerical investigation of the full ETH in physical many-body systems with local interactions by testing the decomposition of higher-order correlators into thermal free cumulants for local operators. We perform exact diagonalization on two classes of local nonintegrable (chaotic) quantum many-body systems: spin chain Hamiltonians and Floquet brickwork unitary circuits. We show that the dynamics of four-time correlation functions are encoded in fourth-order free cumulants, as predicted by ETH. Their dependence on frequency encodes the physical properties of local many-body systems and distinguishes them from structureless, rotationally invariant ensembles of random matrices.

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,

ρ_{AA^*}^{(2)}

,which controls the Rényi-2 reflected entropy. We show that

ρ_{AA^*}^{(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.

The Eigenstate Thermalization Hypothesis (ETH) is a standard tool to understand the thermalization properties of an isolated quantum system. Its generalization to higher order correlations of matrix elements of local operators, dubbed the full ETH, relates the ETH free cumulants to the corresponding thermal free cumulants in the thermodynamic limit. In this work, we numerically test the full ETH predictions using exact diagonalization of two integrable spin models: the Ising and the XXZ Heisenberg models. The differences from the behavior predicted by full ETH in chaotic systems are highlighted and contrasted along the way. We further study the out-of-time-ordered correlator (OTOC) through its approximate decomposition into contributions involving the second- and fourth-order ETH free cumulants. We find that, although the fourth-order contribution governs the late-time behavior of the OTOC in these integrable models, its dynamics differs significantly from that found in nonintegrable systems.

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 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