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arXiv:2602.09087 · Physical review. B./Physical review. B

Eigenstate Thermalization for Local versus Translationally Invariant Observables

Rohit Patil, Marcos Rigol

quant-phcond-mat.stat-mechSubmitted Feb 9, 2026• Announced Feb 11, 2026• 0 citations
v1 · Feb 9, 2026v2 · Sep 9, 2026
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Abstract

Local observables and their translationally invariant counterparts are generally thought as providing the same predictions for experimental measurements. This is used in the context of their expectation values, which are indeed the same in clean systems (up to finite-size effects), but also in the context of their correlation functions, which need not be the same. We examine this intuition from the perspective of the eigenstate thermalization hypothesis. Specifically, we explore the diagonal matrix elements and the spectral functions of local and translationally invariant observables in the spin-1 tilted field Ising chain with periodic and open boundary conditions. We discuss in which ways those observables are different and in which contexts they can be thought as being the same. Furthermore, we unveil a novel form of off-diagonal eigenstate thermalization in translationally invariant systems that applies to pairs of energy eigenstates with different quasimomenta.

Journal reference
Phys. Rev. B 114, L171105 (2026)
DOI
10.1103/fw73-69y5
Comments
7 pages, 7 figures

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Local observables and their translationally invariant counterparts are generally thought as providing the same predictions for experimental measurements. This is used in the context of their expectation values, which are indeed the same in clean systems (up to finite-size effects), but also in the context of their correlation functions, which need not be the same. We examine this intuition from the perspective of the eigenstate thermalization hypothesis. Specifically, we explore the diagonal matrix elements and the spectral functions of local and translationally invariant observables in the spin-1 tilted field Ising chain with periodic and open boundary conditions. We discuss in which ways those observables are different and in which contexts they can be thought as being the same. Furthermore, we unveil a novel form of off-diagonal eigenstate thermalization in translationally invariant systems that applies to pairs of energy eigenstates with different quasimomenta.
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.
We investigate the eigenstate thermalization hypothesis (ETH) for a translationally invariant quantum spin system on the ddd-dimensional cubic lattice under the periodic boundary conditions. It is known that the ETH holds in this model for typical energy eigenstates in the sense that the standard deviation of the expectation values of a local observable in the energy eigenstates within the microcanonical energy shell vanishes in the thermodynamic limit, which is called the weak ETH. Here, it is remarked that the diagonal elements of a local observable in the energy representation shows the large deviation behavior. This result implies that the fraction of atypical eigenstates which do not represent thermal equilibrium is exponentially small.
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.
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.
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.
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.