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.
We analyze a one-dimensional XXZ spin chain in a disordered magnetic field. As the main probes of the system's behavior, we use the sensitivity of eigenstates to adiabatic transformations, as ...expressed through the fidelity susceptibility, in conjunction with the low-frequency asymptotes of the spectral function. We identify a region of maximal chaos-with exponentially enhanced susceptibility-which separates the many-body localized phase from the diffusive ergodic phase. This regime is characterized by slow transport, and we argue that the presence of such slow dynamics highly constrains any possible localization transition in the thermodynamic limit. Rather, the results are more consistent with absence of the localized phase.
We analyze a crossover between ergodic and nonergodic regimes in an interacting spin chain with a dilute density of impurities, defined as spins with a strong local field. The dilute limit allows us ...to unravel some finite size effects and propose a mechanism for the delocalization of these impurities in the thermodynamic limit. In particular, we show that impurities always relax by exchanging energy with the rest of the chain. The relaxation rate only weakly depends on the impurity density and decays exponentially, up to logarithmic corrections, with the field strength. We connect the relaxation to fast operator spreading and show that the same mechanism destabilizes the recursive construction of local integrals of motion at any impurity density. In the high field limit, impurities appear to be localized, and the system is nonergodic, over a wide range of system sizes. However, this is a transient effect, and the eventual delocalization can be understood in terms of a flowing localization length.
We analyze state preparation within a restricted space of local control parameters between adiabatically connected states of control Hamiltonians. We formulate a conjecture that the time integral of ...energy fluctuations over the protocol duration is bounded from below by the geodesic length set by the quantum geometric tensor. The conjecture implies a geometric lower bound for the quantum speed limit (QSL). We prove the conjecture for arbitrary, sufficiently slow protocols using adiabatic perturbation theory and show that the bound is saturated by geodesic protocols, which keep the energy variance constant along the trajectory. Our conjecture implies that any optimal unit-fidelity protocol, even those that drive the system far from equilibrium, are fundamentally constrained by the quantum geometry of adiabatic evolution. When the control space includes all possible couplings, spanning the full Hilbert space, we recover the well-known Mandelstam-Tamm bound. However, using only accessible local controls to anneal in complex models such as glasses or to target individual excited states in quantum chaotic systems, the geometric bound for the quantum speed limit can be exponentially large in the system size due to a diverging geodesic length. We validate our conjecture both analytically by constructing counter-diabatic and fast-forward protocols for a three-level system, and numerically in nonintegrable spin chains and a nonlocal SYK model.
We reveal a continuous dynamical heating transition between a prethermal and an infinite-temperature stage in a clean, chaotic periodically driven classical spin chain. The transition time is a steep ...exponential function of the drive frequency, showing that the exponentially long-lived prethermal plateau, originally observed in quantum Floquet systems, survives the classical limit. Even though there is no straightforward generalization of Floquet's theorem to nonlinear systems, we present strong evidence that the prethermal physics is well described by the inverse-frequency expansion. We relate the stability and robustness of the prethermal plateau to drive-induced synchronization not captured by the expansion. Our results set the pathway to transfer the ideas of Floquet engineering to classical many-body systems, and are directly relevant for photonic crystals and cold atom experiments in the superfluid regime.
From the Aharonov-Bohm effect to general relativity, geometry plays a central role in modern physics. In quantum mechanics, many physical processes depend on the Berry curvature. However, recent ...advances in quantum information theory have highlighted the role of its symmetric counterpart, the quantum metric tensor. In this paper, we perform a detailed analysis of the ground state Riemannian geometry induced by the metric tensor, using the quantum XY chain in a transverse field as our primary example. We focus on a particular geometric invariant, the Gaussian curvature, and show how both integrals of the curvature within a given phase and singularities of the curvature near phase transitions are protected by critical scaling theory. For cases where the curvature is integrable, we show that the integrated curvature provides a new geometric invariant, which like the Chern number characterizes individual phases of matter. For cases where the curvature is singular, we classify three types (integrable, conical, and curvature singularities) and detail situations where each type of singularity should arise. Finally, to connect this abstract geometry to experiment, we discuss three different methods for measuring the metric tensor: via integrating a properly weighted noise spectral function or by using leading-order responses of the work distribution to ramps and quenches in quantum many-body systems.
The ability to prepare a physical system in a desired quantum state is central to many areas of physics such as nuclear magnetic resonance, cold atoms, and quantum computing. Yet, preparing states ...quickly and with high fidelity remains a formidable challenge. In this work, we implement cutting-edge reinforcement learning (RL) techniques and show that their performance is comparable to optimal control methods in the task of finding short, high-fidelity driving protocol from an initial to a target state in nonintegrable many-body quantum systems of interacting qubits. RL methods learn about the underlying physical system solely through a single scalar reward (the fidelity of the resulting state) calculated from numerical simulations of the physical system. We further show that quantum-state manipulation viewed as an optimization problem exhibits a spin-glass-like phase transition in the space of protocols as a function of the protocol duration. Our RL-aided approach helps identify variational protocols with nearly optimal fidelity, even in the glassy phase, where optimal state manipulation is exponentially hard. This study highlights the potential usefulness of RL for applications in out-of-equilibrium quantum physics.
We developed a novel perturbative expansion based on the replica trick for the Floquet Hamiltonian governing the dynamics of periodically kicked systems where the kick strength is the small ...parameter. The expansion is formally equivalent to an infinite resummation of the Baker-Campbell-Hausdorff series in the undriven (nonperturbed) Hamiltonian, while considering terms up to a finite order in the kick strength. As an application of the replica expansion, we analyze an Ising spin 1/2 chain periodically kicked with a magnetic field with a strength h, which has both longitudinal and transverse components. We demonstrate that even away from the regime of high frequency driving, if there is heating, its rate is nonperturbative in the kick strength, bounded from above by a stretched exponential: e^{-const h^{-1/2}}. This guarantees the existence of a very long prethermal regime, where the dynamics is governed by the Floquet Hamiltonian obtained from the replica expansion.
We analyze the problem of optimal adiabatic passage through a quantum critical point. We show that to minimize the number of defects the tuning parameter should be changed as a power law in time. The ...optimal power is proportional to the logarithm of the total passage time multiplied by universal critical exponents characterizing the phase transition. We support our results by the general scaling analysis and by explicit calculations for the transverse-field Ising model.