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arXiv:2405.16975 · Physical Review X

Hydrodynamics and the Eigenstate Thermalization Hypothesis

Luca Capizzi, Jiaozi Wang, Xiansong Xu, Leonardo Mazza, Dario Poletti

Published Mar 14, 2025• 20 citations
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Abstract

The eigenstate thermalization hypothesis (ETH) describes the properties of diagonal and off-diagonal matrix elements of local operators in the eigenenergy basis. In this work, we propose a relation between (i) the singular behavior of the off-diagonal part of ETH at small energy differences and (ii) the smooth profile of the diagonal part of ETH as a function of the energy density. We establish this connection from the decay of the autocorrelation functions of local operators, which is constrained by the presence of local conserved quantities whose evolution is described by hydrodynamics. We corroborate our predictions through numerical simulations of two distinct nonintegrable spin-1 Ising models, exhibiting diffusive and superdiffusive transport behaviors. The simulations are performed using dynamical quantum typicality up to 18 spins, for both infinite- and finite-temperature regimes.

DOI
10.1103/physrevx.15.011059

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The eigenstate thermalization hypothesis (ETH) describes the properties of diagonal and off-diagonal matrix elements of local operators in the eigenenergy basis. In this work, we propose a relation between (i) the singular behavior of the off-diagonal part of ETH at small energy differences and (ii) the smooth profile of the diagonal part of ETH as a function of the energy density. We establish this connection from the decay of the autocorrelation functions of local operators, which is constrained by the presence of local conserved quantities whose evolution is described by hydrodynamics. We corroborate our predictions through numerical simulations of two distinct nonintegrable spin-1 Ising models, exhibiting diffusive and superdiffusive transport behaviors. The simulations are performed using dynamical quantum typicality up to 18 spins, for both infinite- and finite-temperature regimes.
The thermalizing dynamics of many-body systems is often described through the lens of the eigenstate thermalization hypothesis (ETH). ETH postulates that the statistical properties of observables, when expressed in the energy eigenbasis, are described by smooth functions, which also describe correlations among the matrix elements. However, the form of these functions is usually left undetermined, constituting a key missing component of the ETH framework. In this Letter, we investigate the structure of such smooth functions by focusing on their Fourier transform, recently identified as free cumulants. Using nonlinear hydrodynamics, we provide a prediction for the universal scaling of the late-time behavior of time-ordered free cumulants in the thermodynamic limit. The prediction is further corroborated by large-scale numerical simulations of several nonintegrable one-dimensional spin models that exhibit diffusive transport behavior. Good agreement is observed in both infinite and finite-temperature regimes and for a collection of local observables. Our results indicate that the smooth multipoint correlation functions within the ETH framework admit a universal hydrodynamic description at low frequencies.
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 fact that generic isolated many-body quantum systems thermalize is understood using the eigenstate thermalization hypothesis (ETH). In recent years, there has been much interest in the behavior of the ETH spectral functions, which characterize the smooth dependence of the variance of the off-diagonal matrix elements of observables on the associated energy and frequency, and whose low-frequency part contains information about the long-time dynamics. In finite systems described by the ETH, the spectral functions are expected to exhibit plateaus below a characteristic frequency ωETHω^{}_{\mathrm{ETH}}ωETH​. In this regime, the statistics of the matrix elements of observables are expected to be described by random matrix theory. Related frequencies that have been studied in the literature are ωSFFω^{}_{\mathrm{SFF}}ωSFF​, which controls the onset of random-matrix behavior in the spectral form factor, and the transport frequencies ωtrω^{}_{\mathrm{tr}}ωtr​, which are derived from transport coefficients. However, a direct quantitative comparison of these frequencies is lacking. Using exact diagonalization, we conduct such a comparison for the spectral functions of current operators in clean and disordered quantum spin ladders with diffusive energy and spin transport. We find clear evidence for the expected low-frequency plateaus in the ETH spectral functions. For the accessible system sizes, ωSFFω^{}_{\mathrm{SFF}}ωSFF​ is consistent with the extent of the plateaus, while the transport frequencies are systematically larger and lie in the nonuniversal regime of the spectral functions. Our findings highlight the need to better understand the origin of these quantitative differences.
It is commonly believed that quantum isolated systems satisfying the eigenstate thermalization hypothesis (ETH) are diffusive. We show that this assumption is too restrictive since there are systems that are asymptotically in a thermal state yet exhibit anomalous, subdiffusive thermalization. We show that such systems satisfy a modified version of the ETH ansatz and derive a general connection between the scaling of the variance of the off-diagonal matrix elements of local operators, written in the eigenbasis of the Hamiltonian, and the dynamical exponent. We find that for subdiffusively thermalizing systems the variance scales more slowly with system size than expected for diffusive systems. We corroborate our findings by numerically studying the distribution of the coefficients of the eigenfunctions and the off-diagonal matrix elements of local operators of the random field Heisenberg chain, which has anomalous transport in its thermal phase. Surprisingly, this system also has non-Gaussian distributions of the eigenfunctions, thus, directly violating Berry's conjecture.
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.
We investigate the extent to which the eigenstate thermalization hypothesis (ETH) is valid or violated in the nonintegrable and the integrable spin-1/2 XXZ chains. We perform the energy-resolved analysis of statistical properties of matrix elements of observables in the energy eigenstate basis. The Hilbert space is divided into energy shells of constant width, and a block submatrix is constructed whose columns and rows correspond to the eigenstates in the respective energy shells. In each submatrix, we measure the second moment of off-diagonal elements in a column. The columnar second moments are distributed with a finite variance for finite-sized systems. We show that the relative variance of the columnar second moments decreases as the system size increases in the non-integrable system. The self-averaging behavior indicates that the energy eigenstates are statistically equivalent to each other, which is consistent with the ETH. In contrast, the relative variance does not decrease with the system size in the integrable system. The persisting eigenstate-to-eigenstate fluctuation implies that the matrix elements cannot be characterized with the energy parameters only. Our result explains the origin for the breakdown of the fluctuation dissipation theorem in the integrable system. The eigenstate-to-eigenstate fluctuations sheds a new light on the meaning of the ETH.