The eigenstate thermalization hypothesis (ETH) is the leading interpretation in our current understanding of quantum thermalization. Recent results uncovered strong connections between quantum correlations in thermalizing systems and the structure of free probability theory, leading to the notion of full ETH. However, most studies have been performed for ergodic systems and it is still unclear whether or how full ETH manifests in ergodicity-breaking models. We fill this gap by studying standard probes of full ETH in ergodicity-breaking quantum circuits, presenting numerical and analytical results for interacting integrable systems. These probes can display distinct behavior and undergo a different scaling than the ones observed in ergodic systems. For the analytical results we consider an interacting integrable dual-unitary model and present the exact eigenstates, allowing us to analytically express common probes for full ETH. We discuss the underlying mechanisms responsible for these differences and show how the presence of solitons dictates the behavior of ETH-related quantities in the dual-unitary model. We show numerical evidence that this behavior is sufficiently generic away from dual-unitarity when restricted to the appropriate symmetry sectors.
The eigenstate thermalization hypothesis (ETH) is the leading interpretation in our current understanding of quantum thermalization. Recent results uncovered strong connections between quantum correlations in thermalizing systems and the structure of free probability theory, leading to the notion of full ETH. However, most studies have been performed for ergodic systems and it is still unclear whether or how full ETH manifests in ergodicity-breaking models. We fill this gap by studying standard probes of full ETH in ergodicity-breaking quantum circuits, presenting numerical and analytical results for interacting integrable systems. These probes can display distinct behavior and undergo a different scaling than the ones observed in ergodic systems. For the analytical results we consider an interacting integrable dual-unitary model and present the exact eigenstates, allowing us to analytically express common probes for full ETH. We discuss the underlying mechanisms responsible for these differences and show how the presence of solitons dictates the behavior of ETH-related quantities in the dual-unitary model. We show numerical evidence that this behavior is sufficiently generic away from dual-unitarity when restricted to the appropriate symmetry sectors.
The Eigenstate Thermalization Hypothesis (ETH) is a standard tool to understand the thermalization properties of an isolated quantum system. Its generalization to higher order correlations of matrix elements of local operators, dubbed the full ETH, relates the ETH free cumulants to the corresponding thermal free cumulants in the thermodynamic limit. In this work, we numerically test the full ETH predictions using exact diagonalization of two integrable spin models: the Ising and the XXZ Heisenberg models. The differences from the behavior predicted by full ETH in chaotic systems are highlighted and contrasted along the way. We further study the out-of-time-ordered correlator (OTOC) through its approximate decomposition into contributions involving the second- and fourth-order ETH free cumulants. We find that, although the fourth-order contribution governs the late-time behavior of the OTOC in these integrable models, its dynamics differs significantly from that found in nonintegrable systems.
The eigenstate thermalization hypothesis (ETH) has been established as the general framework to understand quantum statistical mechanics. Only recently has the attention been paid to so-called full ETH, which accounts for higher-order correlations among matrix elements, and that can be rationalized theoretically using the language of free probability. In this work, we perform the first numerical investigation of the full ETH in physical many-body systems with local interactions by testing the decomposition of higher-order correlators into thermal free cumulants for local operators. We perform exact diagonalization on two classes of local nonintegrable (chaotic) quantum many-body systems: spin chain Hamiltonians and Floquet brickwork unitary circuits. We show that the dynamics of four-time correlation functions are encoded in fourth-order free cumulants, as predicted by ETH. Their dependence on frequency encodes the physical properties of local many-body systems and distinguishes them from structureless, rotationally invariant ensembles of random matrices.
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.
Eigenstate thermalization hypothesis (ETH) represents a breakthrough in many-body physics since it allows us to link thermalization of physical observables with the applicability of random matrix theory (RMT). Recent years were also extremely fruitful in exploring possible counterexamples to thermalization, ranging, among others, from integrability, single-particle chaos, many-body localization, many-body scars, to Hilbert-space fragmentation. In all these cases the conventional ETH is violated. However, it remains elusive how the conventional ETH breaks down when one approaches the boundaries of ergodicity, and whether the range of validity of the conventional ETH coincides with the validity of RMT-like spectral statistics. Here we bridge this gap and we introduce a scenario of the ETH breakdown in many-body quantum systems, dubbed fading ergodicity regime, which establishes a link between the conventional ETH and nonergodic behavior. We conjecture this scenario to be relevant for the description of finite many-body systems at the boundaries of ergodicity, and we provide numerical and analytical arguments for its validity in the quantum sun model of ergodicity-breaking phase transition. For the latter, we provide evidence that the breakdown of the conventional ETH is not associated with the breakdown of the RMT-like spectral statistics. Published by the American Physical Society 2024
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) 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.