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arXiv:2011.13958 · Physical review. B./Physical review. B

Eigenstate thermalization hypothesis through the lens of autocorrelation functions

Christoph Schönle, D. J. Jansen, Fabian Heidrich‐Meisner, Lev Vidmar

Published Jun 16, 2021• 52 citations
arXivHTMLPDF

Abstract

Matrix elements of observables in eigenstates of generic Hamiltonians are described by the Srednicki ansatz within the eigenstate thermalization hypothesis (ETH). We study a quantum chaotic spin-fermion model in a one-dimensional lattice, which consists of a spin-1/2 XX chain coupled to a single itinerant fermion. In our study, we focus on translationally invariant observables including the charge and energy current, thereby also connecting the ETH with transport properties. Considering observables with a Hilbert-Schmidt norm of one, we first perform a comprehensive analysis of ETH in the model taking into account latest developments. A particular emphasis is on the analysis of the structure of the offdiagonal matrix elements $|\ensuremath{\langle}\ensuremath{\alpha}|\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{O}{|\ensuremath{\beta}\ensuremath{\rangle}|}^{2}$ in the limit of small eigenstate energy differences $\ensuremath{\omega}={E}_{\ensuremath{\beta}}\ensuremath{-}{E}_{\ensuremath{\alpha}}$. Removing the dominant exponential suppression of $|\ensuremath{\langle}\ensuremath{\alpha}|\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{O}{|\ensuremath{\beta}\ensuremath{\rangle}|}^{2}$, we find that (1) the current matrix elements exhibit a system-size dependence that is different from other observables under investigation and (2) matrix elements of several other observables exhibit a Drude-like structure with a Lorentzian frequency dependence. We then show how this information can be extracted from the autocorrelation functions as well. Finally, our study is complemented by a numerical analysis of the fluctuation-dissipation relation for eigenstates in the bulk of the spectrum. We identify the regime of $\ensuremath{\omega}$ in which the well-known fluctuation-dissipation relation is valid with high accuracy for finite systems.

DOI
10.1103/physrevb.103.235137

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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.
We derive the eigenstate thermalization hypothesis (ETH) from a random matrix Hamiltonian by extending the model introduced by Deutsch (1991 Phys. Rev. A 43 2046). We approximate the coupling between a subsystem and a many-body environment by means of a random Gaussian matrix. We show that a common assumption in the analysis of quantum chaotic systems, namely the treatment of eigenstates as independent random vectors, leads to inconsistent results. However, a consistent approach to the ETH can be developed by introducing an interaction between random wave-functions that arises as a result of the orthonormality condition. This approach leads to a consistent form for off-diagonal matrix elements of observables. From there we obtain the scaling of time-averaged fluctuations of generic observables with system size for which we calculate an analytic form in terms of the inverse participation ratio. The analytic results are compared to exact diagonalizations of a quantum spin chain for different physical observables in multiple parameter regimes.
We study the off-diagonal matrix elements of observables that break the translational symmetry of a spin-chain Hamiltonian, and as such connect energy eigenstates from different total quasimomentum sectors. We consider quantum-chaotic and interacting integrable points of the Hamiltonian, and focus on average energies at the center of the spectrum. In the quantum-chaotic model, we find that there is eigenstate thermalization; specifically, the matrix elements are Gaussian distributed with a variance that is a smooth function of ω=E_{α}-E_{β} (E_{α} are the eigenenergies) and scales as 1/D (D is the Hilbert space dimension). In the interacting integrable model, we find that the matrix elements exhibit a skewed log-normal-like distribution and have a variance that is also a smooth function of ω that scales as 1/D. We study in detail the low-frequency behavior of the variance of the matrix elements to unveil the regimes in which it exhibits diffusive or ballistic scaling. We show that in the quantum-chaotic model the behavior of the variance is qualitatively similar for matrix elements that connect eigenstates from the same versus different quasimomentum sectors. We also show that this is not the case in the interacting integrable model for observables whose translationally invariant counterpart does not break integrability if added as a perturbation to the Hamiltonian.
The Eigenstate Thermalization Hypothesis (ETH) explains how isolated quantum many-body systems thermalize by proposing that each energy eigenstate is already thermal. The hypothesis is believed to be essential to the understanding of quantum chaos and implies various important thermodynamic relations. While by now there are many numerical validations of the ETH, only few analytical arguments supporting the ETH have been found (M. Rigol et al., Nature 452, 2008). One of them is based on a semiclassical limit (M. Srednicki, Phys. Rev. E 50, 1994). Another important reasoning was given by Deutsch, who showed how a small interaction, which is modeled by a random matrix, leads to ETH (J.M. Deutsch, Phys. Rev. A 43, 1991). This thesis aims to find justification for the assumptions made by Deutsch from a microscopic point of view. The idea is to analyze whether a generic quantum system can be brought to a form that satisfies Deutsch's ansatz. The method is to use flow equations to define unitary transformations to map the initial Hamiltonian to an effective one, which can be compared to a random matrix. The test system is a one-dimensional hardcore boson model, which is set up using exact diagonalization and numerous symmetries. We analyze the statistical properties of the numerically obtained Hamiltonian matrices and test for normality and compare them to a random transformation. In a second step we verify that the continuous unitary transformations do not destroy the structure of typical few-body observables by studying the scaling of the inverse participation ratio. The obtained results support our hypothesis that it is possible to extend Deutsch's argument to generic microscopic many-body Hamiltonians.
Despite the unitary evolution of closed quantum systems, long-time expectation of local observables are well described by thermal ensembles, providing the foundation of quantum statistical mechanics. A promising route to understanding this quantum thermalization is the eigenstate thermalization hypothesis (ETH), which posits that individual energy eigenstates already appear locally thermal. Subsequent studies have extended this concept to the full ETH, which captures higher-order correlations among matrix elements through nontrivial relations. In this work, we perform a detailed exact-diagonalization study of finite-size corrections to these relations in the canonical ensemble. We distinguish two distinct sources of corrections: those arising from energy fluctuations, which decay polynomially with system size, and those originating from fluctuations within each energy window, which decay exponentially with system size. In particular, our analysis resolves the puzzle that, for certain observables, finite-size corrections exhibit anomalous growth with increasing system size even in chaotic systems. Our results provide a systematic and practical methodology for validating the full ETH in quantum many-body systems.
We derive the eigenstate thermalization hypothesis (ETH) from a random matrix Hamiltonian by extending the model introduced by Deutsch (1991 Phys. Rev. A 43 2046). We approximate the coupling between a subsystem and a many-body environment by means of a random Gaussian matrix. We show that a common assumption in the analysis of quantum chaotic systems, namely the treatment of eigenstates as independent random vectors, leads to inconsistent results. However, a consistent approach to the ETH can be developed by introducing an interaction between random wave-functions that arises as a result of the orthonormality condition. This approach leads to a consistent form for off-diagonal matrix elements of observables. From there we obtain the scaling of time-averaged fluctuations of generic observables with system size for which we calculate an analytic form in terms of the inverse participation ratio. The analytic results are compared to exact diagonalizations of a quantum spin chain for different physical observables in multiple parameter regimes.