We study thermalization in a disordered one-dimensional interacting bosonic system described by the Aubry-Andre model using full exact diagonalization. We find a broad chaotic energy window where the system's eigenstates satisfy the Eigenstate Thermalization Hypothesis (ETH), demonstrated by the smooth energy dependence of observables like entanglement entropy and local particle number, whose fluctuations decrease with system size. Dynamically, we investigate the equilibration of initial Fock states and find that thermalization is not universal. The key finding is a direct and nontrivial correlation between an initial state's delocalization in the energy eigenbasis quantified by the Participation Ratio (PR) and its subsequent equilibration. States with a high PR consistently evolve toward the microcanonical ensemble prediction, whereas those exhibiting a low PR display deviations whose magnitude inversely correlates with the PR value. This connection is quantitatively confirmed by the trace distance, providing a powerful, experimentally relevant diagnostic for predicting which initial states will reach thermal equilibrium.
We study thermalization in a disordered one-dimensional interacting bosonic system described by the Aubry-Andre model using full exact diagonalization. We find a broad chaotic energy window where the system's eigenstates satisfy the Eigenstate Thermalization Hypothesis (ETH), demonstrated by the smooth energy dependence of observables like entanglement entropy and local particle number, whose fluctuations decrease with system size. Dynamically, we investigate the equilibration of initial Fock states and find that thermalization is not universal. The key finding is a direct and nontrivial correlation between an initial state's delocalization in the energy eigenbasis quantified by the Participation Ratio (PR) and its subsequent equilibration. States with a high PR consistently evolve toward the microcanonical ensemble prediction, whereas those exhibiting a low PR display deviations whose magnitude inversely correlates with the PR value. This connection is quantitatively confirmed by the trace distance, providing a powerful, experimentally relevant diagnostic for predicting which initial states will reach thermal equilibrium.
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.
We investigate the onset of "eigenstate thermalization" and the crossover to ergodicity in a system of one-dimensional fermions with increasing interaction. We analyze the fluctuations in the expectation values of most relevant few-body operators with respect to eigenstates. It turns out that these are intimately related to the inverse participation ratio of eigenstates displayed in the operator eigenbasis. Based on this observation, we find good evidence that eigenstate thermalization should set in even for vanishingly small perturbations in the thermodynamic limit.
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.
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.
A plausible mechanism of thermalization in isolated quantum systems is based on the strong version of the eigenstate thermalization hypothesis (ETH), which states that all the energy eigenstates in the microcanonical energy shell have thermal properties. We numerically investigate the ETH by focusing on the large deviation property, which directly evaluates the ratio of athermal energy eigenstates in the energy shell. As a consequence, we have systematically confirmed that the strong ETH is indeed true even for near-integrable systems. Furthermore, we found that the finite-size scaling of the ratio of athermal eigenstates is a double exponential for nonintegrable systems. Our result illuminates the universal behavior of quantum chaos, and suggests that a large deviation analysis would serve as a powerful method to investigate thermalization in the presence of the large finite-size effect.
We study whether a generic isolated quantum system initially set out of equilibrium can be considered as localized close to its initial state. Our approach considers the time evolution in the Krylov basis, which maps the system's dynamics onto that of a particle moving in a one-dimensional lattice where both the energy in the lattice sites and the tunneling from one lattice site to the next are inhomogeneous. By tying the dynamical propagation in the Krylov basis to that in the basis of microstates, we infer qualitative criteria that allow distinguishing systems that remain localized close to their initial state from systems that undergo quantum thermalization. These criteria are system-dependent and involve the expectation values and standard deviations of both the coupling strengths between Krylov states and their energy. We verify their validity by inspecting two cases: Anderson localization as a function of dimension and the out-of-equilibrium dynamics of a many-body dipolar spin system. We finally investigate the Wigner surmise and the eigenstate thermalization hypothesis that have both been proposed to characterize quantum chaotic systems. We show that when the average value of the nondiagonal terms in the Lanczos matrix is large compared to their fluctuations and to the fluctuations of the energy expectation values, which typically corresponds to delocalized quantum systems according to our criteria, there can be level repulsion of eigenenergies (also known as spectral rigidity), similar to that of the Wigner-Dyson statistics of energy levels; and we also demonstrate that in the same regime, the expectation values of typical local observables only weakly vary as a function of eigenstates, an important condition for the eigenstate thermalization hypothesis. Our demonstration assumes that, in the chaotic regime, the observable is sufficiently diagonal in the basis of Krylov states. Published by the American Physical Society 2024