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arXiv:2008.09318 · Physical review. E

Eigenstate thermalization hypothesis and eigenstate-to-eigenstate fluctuations

Jae Dong Noh

Published Jan 25, 2021• 17 citations
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Abstract

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.

DOI
10.1103/physreve.103.012129

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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.
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.
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.
The Eigenstate Thermalization Hypothesis explains thermalization in isolated quantum systems through the statistical properties of observables in the energy eigenbasis. We investigate the crossover from integrability to chaos in the spin-1/21/21/2 XXZ chain, establishing a direct correspondence between the spectral correlations of the Hamiltonian and local observables expressed in the energy eigenbasis as a signature of ergodicity breaking. By introducing a local perturbation that drives the system from integrability to chaos, we track the standard ETH indicators and the eigenstate entanglement entropy. We introduce a submatrix-based framework for analyzing local observables in the energy eigenbasis. By extracting real-symmetric blocks along the diagonal of the local observables represented in eigenbasis, we show that these submatrices exhibit both the short-range and long-range spectral features of the Hamiltonian. Remarkably, this correspondence persists even in a partially ergodic regime, indicating that the emergence of chaos is already encoded locally within the observables' matrix structure and that small blocks are sufficient to capture the underlying spectral correlations.
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.
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 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.