This work aims at understanding the interplay between the eigenstate thermalization hypothesis (ETH), initial state independent equilibration, and quantum chaos in systems that do not have a direct classical counterpart. It is based on numerical investigations of asymmetric Heisenberg spin ladders with varied interaction strengths between the legs, i.e., along the rungs. The relaxation of the energy difference between the legs is investigated. Two different parameters, both intended to quantify the degree of accordance with the ETH, are computed. Both indicate violation of the ETH at large interaction strengths but at different thresholds. Indeed, the energy difference is found not to relax independently of its initial value above some critical interaction strength, which coincides with one of the thresholds. At the same point the level statistics shift from Poisson-type to Wigner-type. Hence, the system may be considered to become integrable again in the strong interaction limit.
This work aims at understanding the interplay between the eigenstate thermalization hypothesis (ETH), initial state independent equilibration, and quantum chaos in systems that do not have a direct classical counterpart. It is based on numerical investigations of asymmetric Heisenberg spin ladders with varied interaction strengths between the legs, i.e., along the rungs. The relaxation of the energy difference between the legs is investigated. Two different parameters, both intended to quantify the degree of accordance with the ETH, are computed. Both indicate violation of the ETH at large interaction strengths but at different thresholds. Indeed, the energy difference is found not to relax independently of its initial value above some critical interaction strength, which coincides with one of the thresholds. At the same point the level statistics shift from Poisson-type to Wigner-type. Hence, the system may be considered to become integrable again in the strong interaction limit.
We study the validity of the eigenstate thermalization hypothesis (ETH) and its role for the occurrence of initial-state independent (ISI) equilibration in closed quantum many-body systems. Using the concept of dynamical typicality, we present an extensive numerical analysis of energy exchange in integrable and nonintegrable spin-1/2 systems of large size outside the range of exact diagonalization. In the case of nonintegrable systems, our finite-size scaling shows that the ETH becomes valid in the thermodynamic limit and can serve as the underlying mechanism for ISI equilibration. In the case of integrable systems, however, indication of ISI equilibration has been observed despite the violation of the ETH. We establish a connection between this observation and the need of choosing a proper parameter within the ETH.
The eigenstate thermalization hypothesis (ETH) is a successful theory that establishes the criteria for ergodicity and thermalization in isolated quantum many-body systems. In this work, we investigate the thermalization properties of a spin-1/2 XXZ chain with linearly inhomogeneous interactions. We demonstrate that introduction of the inhomogeneous interactions leads to an onset of quantum chaos and thermalization, which, however, becomes inhibited for sufficiently strong inhomogeneity. To exhibit ETH, and to display its breakdown upon varying the strength of interactions, we probe statistics of energy levels and properties of matrix elements of local observables in eigenstates of the inhomogeneous XXZ spin chain. Moreover, we investigate the dynamics of the entanglement entropy and the survival probability which further evidence the thermalization and its breakdown in the considered model. We outline a way to experimentally realize the XXZ chain with linearly inhomogeneous interactions in systems of ultracold atoms. Our results highlight a mechanism of emergence of ETH due to insertion of inhomogeneities in an otherwise integrable system and illustrate the arrest of quantum dynamics in the presence of strong interactions.
The eigenstate thermalization hypothesis (ETH) is a successful theory that establishes the criteria for ergodicity and thermalization in isolated quantum many-body systems. In this work, we investigate the thermalization properties of spin-$ 1/2 $ XXZ chain with linearly-inhomogeneous interactions. We demonstrate that introduction of the inhomogeneous interactions leads to an onset of quantum chaos and thermalization, which, however, becomes inhibited for sufficiently strong inhomogeneity. To exhibit ETH, and to display its breakdown upon varying the strength of interactions, we probe statistics of energy levels and properties of matrix elements of local observables in eigenstates of the inhomogeneous XXZ spin chain. Moreover, we investigate the dynamics of the entanglement entropy and the survival probability which further evidence the thermalization and its breakdown in the considered model. We outline a way to experimentally realize the XXZ chain with linearly-inhomogeneous interactions in systems of ultracold atoms. Our results highlight a mechanism of emergence of ETH due to insertion of inhomogeneities in an otherwise integrable system and illustrate the arrest of quantum dynamics in presence of strong interactions.
The emergence of statistical mechanics for isolated classical systems comes about through chaotic dynamics and ergodicity. Here we review how similar questions can be answered in quantum systems. The crucial point is that individual energy eigenstates behave in many ways like a statistical ensemble. A more detailed statement of this is named the eigenstate thermalization hypothesis (ETH). The reasons for why it works in so many cases are rooted in the early work of Wigner on random matrix theory and our understanding of quantum chaos. The ETH has now been studied extensively by both analytic and numerical means, and applied to a number of physical situations ranging from black hole physics to condensed matter systems. It has recently become the focus of a number of experiments in highly isolated systems. Current theoretical work also focuses on where the ETH breaks down leading to new interesting phenomena. This review of the ETH takes a somewhat intuitive approach as to why it works and how this informs our understanding of many body quantum states.
In classical systems, chaos is clearly defined via the behavior of trajectories. In quantum systems with a classical analogue one finds that the transition from regular to chaotic dynamics is signified by a change in the spectral statistics. This has been found to remain true for quantum systems with no classical analogue, including many-body systems. Furthermore, quantum chaotic systems explore all the allowed configurations in the Hilbert space, i.e. they are ergodic, while integrable systems, and systems in the many-body localized phase, are restricted to a certain subspace of the available phase space, and hence strongly break ergodicity. In this dissertation, we study the intermediate behavior between ergodicity and localization, i.e. the weak breaking of ergodicity. The model examined is the PXP spin chain model, where spins are allowed to flip only under certain kinetic constraints. We start by reproducing some already established results. First, we explore the eigenstate thermalization hypothesis (ETH) for this model and demonstrate the existence of a small number of states, throughout the PXP spectrum, that violate the ETH. Then we study the level-spacing statistics of the model, a well-known quantum chaos diagnostic, which turns out to be close to semi-Poisson and approach Wigner--Dyson statistics for large system sizes. Moreover, we examine various aspects of the model that have not been studied before. For example, the eigenvector component statistics, another quantum chaos diagnostic, for the PXP model turn out to be non-Gaussian. Finally, we perform a quench, in order to study how the energy spreads throughout the system, and observe ballistic fronts.
This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the implications of quantum chaos and ETH to thermodynamics. Basic thermodynamic relations are derived, including the second law of thermodynamics, the fundamental thermodynamic relation, fluctuation theorems, the fluctuation–dissipation relation, and the Einstein and Onsager relations. In particular, it is shown that quantum chaos allows one to prove these relations for individual Hamiltonian eigenstates and thus extend them to arbitrary stationary statistical ensembles. In some cases, it is possible to extend their regimes of applicability beyond the standard thermal equilibrium domain. We then show how one can use these relations to obtain nontrivial universal energy distributions in continuously driven systems. At the end of the review, we briefly discuss the relaxation dynamics and description after relaxation of integrable quantum systems, for which ETH is violated. We present results from numerical experiments and analytical studies of quantum quenches at integrability. We introduce the concept of the generalized Gibbs ensemble and discuss its connection with ideas of prethermalization in weakly interacting systems.