The recent discovery that for large Hilbert spaces, almost all (that is, typical) Hamiltonians have eigenstates that place small subsystems in thermal equilibrium, has shed much light on the origins of irreversibility and thermalization. Here we give numerical evidence that many-body lattice systems generically approach typicality as the number of subsystems is increased, and thus provide further support for the eigenstate thermalization hypothesis. Our results indicate that the deviation of many-body systems from typicality decreases exponentially with the number of systems. Further, by averaging over a number of randomly selected nearest-neighbor interactions, we obtain a powerlaw for the atypicality as a function of the Hilbert space dimension, distinct from the power law possessed by random Hamiltonians.
The recent discovery that for large Hilbert spaces, almost all (that is, typical) Hamiltonians have eigenstates that place small subsystems in thermal equilibrium, has shed much light on the origins of irreversibility and thermalization. Here we give numerical evidence that many-body lattice systems generically approach typicality as the number of subsystems is increased, and thus provide further support for the eigenstate thermalization hypothesis. Our results indicate that the deviation of many-body systems from typicality decreases exponentially with the number of systems. Further, by averaging over a number of randomly selected nearest-neighbor interactions, we obtain a powerlaw for the atypicality as a function of the Hilbert space dimension, distinct from the power law possessed by random Hamiltonians.
The eigenstate thermalization hypothesis (ETH), which dictates that all diagonal matrix elements within a small energy shell be almost equal, is a major candidate to explain thermalization in isolated quantum systems. According to the typicality argument, the maximum variations of such matrix elements should decrease exponentially with increasing the size of the system, which implies the ETH. We show, however, that the typicality argument does not apply to most few-body observables for few-body Hamiltonians when the width of the energy shell decreases at most polynomially with increasing the size of the system.
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.
The eigenstate thermalization hypothesis (ETH) provides a powerful framework for understanding thermalization in isolated quantum many-body systems, yet a complete and conceptually transparent derivation has remained elusive. In this work, we derive the structure of ETH from a minimal dynamical principle, which we term the eigenstate typicality principle (ETP), together with general kinematic ingredients arising from entropy maximization, Hilbert-space geometry, and locality. ETP asserts that in quantum-chaotic systems, energy eigenstates are statistically indistinguishable, with respect to local measurements, from states drawn from the Haar measure on a narrow microcanonical shell. Within this framework, diagonal ETH arises from concentration of measure, provided that eigenstate typicality holds. The structure of off-diagonal matrix elements is then fixed by entropic scaling and the finite-time dynamical correlations of local observables, with ETP serving as the dynamical bridge to energy eigenstates, without invoking random-matrix assumptions. Our results establish ETH as a consequence of entropy, Hilbert-space geometry, and chaos-induced eigenstate typicality, and clarify its regime of validity across generic quantum-chaotic many-body systems, thereby deepening our understanding of quantum thermalization and the emergence of statistical mechanics from unitary many-body dynamics.
According to the eigenstate thermalization hypothesis (ETH), the eigenstate-to-eigenstate fluctuations of expectation values of local observables should decrease with increasing system size. In approaching the thermodynamic limit-the number of sites and the particle number increasing at the same rate-the fluctuations should scale as ∼D^{-1/2} with the Hilbert space dimension D. Here, we study a different limit-the classical or semiclassical limit-by increasing the particle number in fixed lattice topologies. We focus on the paradigmatic Bose-Hubbard system, which is quantum-chaotic for large lattices and shows mixed behavior for small lattices. We derive expressions for the expected scaling, assuming ideal eigenstates having Gaussian-distributed random components. We show numerically that, for larger lattices, ETH scaling of physical midspectrum eigenstates follows the ideal (Gaussian) expectation, but for smaller lattices, the scaling occurs via a different exponent. We examine several plausible mechanisms for this anomalous scaling.
We provide a pedagogical introduction to eigenstate thermalization. This phenomenon, which occurs in generic quantum many-body systems, allows one to understand why thermalization takes place in isolated systems under unitary dynamics. We motivate eigenstate thermalization using random matrix theory and discuss recent complementary results for the volume-law entanglement entropy of Haar-random states. We discuss numerical results that highlight the corresponding behaviors in quantum many-body systems.
Using the ergodicity principle for the expectation values of several types of observables, we investigate the thermalization process in isolated fermionic systems.These are described by the two-body random ensemble, which is a paradigmatic model to study quantum chaos and especially the dynamical transition from integrability to chaos.By means of exact diagonalizations we analyze the relevance of the eigenstate thermalization hypothesis as well as the influence of other factors, such as the energy and structure of the initial state, or the dimension of the Hilbert space.We also obtain analytical expressions linking the degree of thermalization for a given observable with the so-called number of principal components for transition strengths originating at a given energy, with the dimensions of the whole Hilbert space and microcanonical energy shell, and with the correlations generated by the observable.As the strength of the residual interaction is increased, an order-to-chaos transition takes place, and we show that the onset of Wigner spectral fluctuations, which is the standard signature of chaos, is not sufficient to guarantee thermalization in finite systems.When all the signatures of chaos are fulfilled, including the quasicomplete delocalization of eigenfunctions, the eigenstate thermalization hypothesis is the mechanism responsible for the thermalization of certain types of observables, such as (linear combinations of) occupancies and strength function operators.Our results also suggest that fully chaotic systems will thermalize relative to most observables in the thermodynamic limit.