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arXiv:2610.30070

A random sampling method to test the weak Eigenstate Thermalization Hypothesis

Nicolaus A. Chlanda, John Keim, Isabel Martinez-Robles, Sarah E. Spielman, Thomas J. Carroll, Michael W. Noel

quant-phSubmitted Oct 21, 2026• Announced Oct 22, 2026• 0 citations
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Abstract

We introduce a random sampling method for testing the weak Eigenstate Thermalization Hypothesis (wETH) that is particularly useful for large, sparse Hamiltonians. We test the method on a well-studied Rydberg atom system that exhibits both two- and three-body dipole-dipole interactions. Our results show that our method allows for the study of system sizes significantly beyond the limits of exact diagonalization. We find that the fluctuations in eigenstate expectation values decay as the Hilbert space dimension $D^{-1/2}$, consistent with wETH and in agreement with previous numerical results.

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We introduce a random sampling method for testing the weak Eigenstate Thermalization Hypothesis (wETH) that is particularly useful for large, sparse Hamiltonians. We test the method on a well-studied Rydberg atom system that exhibits both two- and three-body dipole-dipole interactions. Our results show that our method allows for the study of system sizes significantly beyond the limits of exact diagonalization. We find that the fluctuations in eigenstate expectation values decay as the Hilbert space dimension D−1/2D^{-1/2}D−1/2, consistent with wETH and in agreement with previous numerical results.
Isolated quantum systems are typically observed to come to thermal equilibrium. The weak Eigenstate Thermalization Hypothesis (wETH) states that the vast majority of eigenvectors are thermal. This implies that most initial states should thermalize. Studying the rare cases in which they fail to thermalize could reveal insight into quantum dynamics. In our experiment, we excite Rydberg atoms and allow them to exchange energy via dipole-dipole interactions. We find that they do not thermalize. We present numerical and theoretical results that show that our experimental initial state is not typical.
We consider the set of all initial states within a microcanonical energy shell of an isolated many-body quantum system, which exhibit an arbitrary but fixed nonequilibrium expectation value for some given observable A. On the condition that this set is not too small, it is shown by means of a dynamical typicality approach that most such initial states exhibit thermalization if and only if A satisfies the so-called weak eigenstate thermalization hypothesis (wETH). Here, thermalization means that the expectation value of A spends most of its time close to the microcanonical value after initial transients have died out. The wETH means that, within the energy shell, most eigenstates of the pertinent system Hamiltonian exhibit very similar expectation values of A.
According to the eigenstate thermalization hypothesis (ETH), even isolated quantum systems can thermalize because the eigenstate-to-eigenstate fluctuations of typical observables vanish in the limit of large systems. Of course, isolated systems are by nature finite and the main way of computing such quantities is through numerical evaluation for finite-size systems. Therefore, the finite-size scaling of the fluctuations of eigenstate expectation values is a central aspect of the ETH. In this work, we present numerical evidence that for generic nonintegrable systems these fluctuations scale with a universal power law D-1/2 with the dimension D of the Hilbert space. We provide heuristic arguments, in the same spirit as the ETH, to explain this universal result. Our results are based on the analysis of three families of models and several observables for each model. Each family includes integrable members and we show how the system size where the universal power law becomes visible is affected by the proximity to integrability.
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.
Motivated by recent ion experiments on tunable long-range interacting quantum systems [Neyenhuis et al., Sci. Adv. 3, e1700672 (2017)SACDAF2375-254810.1126/sciadv.1700672], we test the strong eigenstate thermalization hypothesis for systems with power-law interactions ∼1/r^{α}. We numerically demonstrate that the strong eigenstate thermalization hypothesis typically holds, at least for systems with α≥0.6, which include Coulomb, monopole-dipole, and dipole-dipole interactions. Compared with short-range interacting systems, the eigenstate expectation value of a generic local observable is shown to deviate significantly from its microcanonical ensemble average for long-range interacting systems. We find that Srednicki's ansatz breaks down for α≲1.0, at least for relatively large system sizes.
We propose and numerically validate an efficient single-eigenstate diagnostic for the eigenstate thermalization hypothesis (ETH) based on a perturbed eigenstate quench protocol. By introducing a weak random perturbation to an energy eigenstate to break its stationarity, we characterize the time-averaged subsystem evolution speed as a function of the subsystem-to-total system size ratio. The diagnostic relies on a robust qualitative distinction: eigenstates satisfying ETH exhibit an S-shaped curve with a clear inflection point near half the system size, while ETH-violating eigenstates display a convex J-shaped profile. We benchmark the criterion across paradigmatic one-dimensional spin chains covering chaotic, integrable, many-body localized, and quantum many-body scar regimes, obtaining full agreement with established thermalization phenomenology. Our method circumvents the need for explicit thermal ensemble construction, providing a robust, experimentally feasible probe of eigenstate thermalization at the single-eigenstate level.