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arXiv:1906.01522 · Physical Review Letters

Eigenstate Thermalization and Rotational Invariance in Ergodic Quantum Systems

Laura Foini, Jorge Kurchan B

Published Dec 23, 2019• 38 citations
arXivHTMLPDF

Abstract

Generic rotationally invariant matrix models satisfy a simple relation: the probability distribution of half the difference between any two diagonal elements and the one of off-diagonal elements are the same. In the spirit of the eigenstate thermalization hypothesis, we test the hypothesis that the same relation holds in quantum systems that are nonlocalized, when one considers small energy differences. The relation provides a stringent test of the eigenstate thermalization hypothesis beyond the Gaussian ensemble. We apply it to a disordered spin chain, the Sachdev-Ye-Kitaev model, and a Floquet system.

DOI
10.1103/physrevlett.123.260601

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Generic rotationally invariant matrix models satisfy a simple relation: the probability distribution of half the difference between any two diagonal elements and the one of off-diagonal elements are the same. In the spirit of the eigenstate thermalization hypothesis, we test the hypothesis that the same relation holds in quantum systems that are nonlocalized, when one considers small energy differences. The relation provides a stringent test of the eigenstate thermalization hypothesis beyond the Gaussian ensemble. We apply it to a disordered spin chain, the Sachdev-Ye-Kitaev model, and a Floquet system.
We verify that the eigenstate thermalization hypothesis (ETH) holds universally for locally interacting quantum many-body systems. Introducing random matrix ensembles with interactions, we numerically obtain a distribution of maximum fluctuations of eigenstate expectation values for different realizations of interactions. This distribution, which cannot be obtained from the conventional random matrix theory involving nonlocal correlations, demonstrates that an overwhelming majority of pairs of local Hamiltonians and observables satisfy the ETH with exponentially small fluctuations. The ergodicity of our random matrix ensembles breaks down because of locality.
The Eigenstate Thermalization Hypothesis (ETH) was developed as a framework for understanding how the principles of statistical mechanics emerge in the long-time limit of isolated quantum many-body systems. Since then, ETH has shifted the attention towards the study of matrix elements of physical observables in the energy eigenbasis. In this work, we revisit recent developments leading to the formulation of full ETH, a generalization of the original ETH ansatz that accounts for multi-point correlation functions. Using tools from free probability, we explore the implications of local rotational invariance, a property that emerges from the statistical invariance of observables under random basis transformations induced by small perturbations of the Hamiltonian. This approach allows us to make quantitative predictions and derive an analytical characterization of subleading corrections to matrix-element correlations, thereby refining the ETH ansatz. Moreover, our analysis links the statistical properties of matrix elements under random basis changes to the empirical averages over energy windows that are usually considered when dealing with a single instance of the ensemble. We validate our analytical predictions through comparison with numerical simulations in non-integrable Floquet systems.
The eigenstate thermalization hypothesis (ETH) was developed as a framework for understanding how the principles of statistical mechanics emerge in the long-time limit of isolated quantum many-body systems. Since then, ETH has shifted the attention toward the study of matrix elements of physical observables in the energy eigenbasis. In this work, we revisit recent developments leading to the formulation of full ETH, a generalization of the original ETH ansatz that accounts for multi-point correlation functions. Using tools from free probability, we explore the implications of local rotational invariance, a property that emerges from the statistical invariance of observables under random basis transformations induced by small perturbations of the Hamiltonian. This approach allows us to make quantitative predictions and derive an analytical characterization of subleading corrections to matrix-element correlations, thereby refining the ETH ansatz. Moreover, our analysis links the statistical properties of matrix elements under random basis changes to the empirical averages over energy windows that are usually considered when dealing with a single instance of the ensemble. We validate our analytical predictions through comparison with numerical simulations in non-integrable Floquet systems.
The thermalization phenomenon and many-body quantum statistical properties are studied on the example of several observables in isolated spin-chain systems, both integrable and generic nonintegrable. While diagonal matrix elements for nonintegrable models comply with the eigenstate thermalization hypothesis, the integrable systems show evident deviations and similarity to properties of noninteracting many-fermion models. The finite-size scaling reveals that the crossover between the two regimes is given by a scale closely related to the scattering length. Low-frequency off-diagonal matrix elements related to dc transport quantities also follow in a generic system a behavior analogous to the eigenstate thermalization hypothesis, however, unrelated to one of the diagonal matrix elements.
Using numerical exact diagonalization, we study matrix elements of a local spin operator in the eigenbasis of two different nonintegrable quantum spin chains. Our emphasis is on the question to what extent local operators can be represented as random matrices and, in particular, to what extent matrix elements can be considered as uncorrelated. As a main result, we show that the eigenvalue distribution of band submatrices at a fixed energy density is a sensitive probe of the correlations between matrix elements. We find that, on the scales where the matrix elements are in a good agreement with all standard indicators of the eigenstate thermalization hypothesis, the eigenvalue distribution still exhibits clear signatures of the original operator, implying correlations between matrix elements. Moreover, we demonstrate that at much smaller energy scales, the eigenvalue distribution approximately assumes the universal semicircle shape, indicating transition to the random-matrix behavior, and in particular that matrix elements become uncorrelated.
Local observables and their translationally invariant counterparts are generally thought to provide the same predictions for experiments. While this equivalence holds for expectation values in clean systems (up to finite-size effects), it is often assumed to extend to correlation functions, where it need not hold. We examine this assumption from the perspective of the eigenstate thermalization hypothesis. Specifically, we explore the spectral functions of local and translationally invariant observables in the spin-1 tilted-field Ising chain with periodic and open boundary conditions. We identify the contexts in which these observables and boundary conditions differ and those in which they are interchangeable. We unveil an off-diagonal eigenstate thermalization in translationally invariant systems for matrix elements between energy eigenstates with different quasimomenta.