Bridging the gap between macroscopic thermodynamics and microscopic quantum dynamics is one of the most fundamental problems in statistical physics, dating back to von Neumann.Recently, nonequilibrium dynamics of isolated quantum many-body systems have been observed in highly controllable quantum systems, such as ultracold atoms, superconducting qubits, and trapped ions.Both theoretical and experimental studies have revealed that isolated manybody systems thermalize under the unitary time evolution.These studies triggered progress in understanding mechanisms of thermalization.A widely accepted mechanism of thermalization is based on a conjecture called the eigenstate thermalization hypothesis (ETH).The ETH is a counterpart of ergodicity in quantum systems and states that all the energy eigenstates of a chaotic Hamiltonian are indistinguishable from the thermal states.The ETH has been numerically confirmed for many nonintegrable systems, while it is violated in integrable systems and localized systems.Another essential concept of thermodynamics is the second law, which is formulated in several ways, such as the entropy increase and the impossibility of the perpetual motion of the second kind.The second law in quantum systems has been studied based on quantum * 1 The Srednicki's ansatz (2.29) is not a mathematically rigorous statement, because the left-hand side of Eq. (2.29) is not a stochastic variable, while the right-hand side is a stochastic variable.
Bridging the gap between macroscopic thermodynamics and microscopic quantum dynamics is one of the most fundamental problems in statistical physics, dating back to von Neumann.Recently, nonequilibrium dynamics of isolated quantum many-body systems have been observed in highly controllable quantum systems, such as ultracold atoms, superconducting qubits, and trapped ions.Both theoretical and experimental studies have revealed that isolated manybody systems thermalize under the unitary time evolution.These studies triggered progress in understanding mechanisms of thermalization.A widely accepted mechanism of thermalization is based on a conjecture called the eigenstate thermalization hypothesis (ETH).The ETH is a counterpart of ergodicity in quantum systems and states that all the energy eigenstates of a chaotic Hamiltonian are indistinguishable from the thermal states.The ETH has been numerically confirmed for many nonintegrable systems, while it is violated in integrable systems and localized systems.Another essential concept of thermodynamics is the second law, which is formulated in several ways, such as the entropy increase and the impossibility of the perpetual motion of the second kind.The second law in quantum systems has been studied based on quantum * 1 The Srednicki's ansatz (2.29) is not a mathematically rigorous statement, because the left-hand side of Eq. (2.29) is not a stochastic variable, while the right-hand side is a stochastic variable.
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.
In this Thesis, we investigate problems of thermalization in closed (or isolated) and open quantum many-body systems, which are deeply tied to the foundations of quantum statistical mechanics.Such studies have attracted growing interest thanks to recent experimental realizations of controllable quantum many-body systems, e.g., cold atoms and ions.We firstly discuss thermalization in closed quantum many-body systems.The eigenstate thermalization hypothesis (ETH) has been proposed as a sufficient condition for a system to be described by the microcanonical ensemble after a long-time dynamics.The ETH is expected to hold true for generic systems except for some systems such as many-body localized (MBL) systems with strong disorder.On the other hand, the reason why the ETH generically holds truehas not yet been fully understood.We here rigorously show that the typicality argument, which has long been thought to justify the ETH, does not hold true for a realistic setup with few-body properties of observables and Hamiltonians.We secondly discuss thermalization in open quantum many-body systems.State-of-the-art experiments (e.g., experiments under continuous measurement) have now enabled us to control dissipation, which may enrich properties of closed systems.As one of the fundamental problems of thermalization in open quantum systems, we study the MBL in non-Hermitian systems, which are relevant for continuously measured systems.We show that the transition between delocalized (ETH) and MBL phases occurs even for non-Hermitian many-body Hamiltonians and affects spectral and dynamical properties.In particular, we find that a novel real-complex transition occurs upon MBL and affects the dynamical stability in non-Hermitian interacting systems with asymmetric hopping.Just as delocalized phases in closed systems are characterized by Hermitian random matrices, those in open quantum systems are characterized by non-Hermitian random matrices.On the other hand, in contrast with the Hermitian cases, the universality in non-Hermitian random matrices and its relation to symmetries are not well understood.In Hermitian random matrices, there are three universality classes of local spectral statistics dependent on time-reversal symmetry (TRS).However, TRS is known not to alter universality in non-Hermitian random matrices.We here show that three different universal level-spacing distributions appear even for non-Hermitian random matrices when we consider transposition symmetry instead of TRS.This result serves as a basis for characterizing nonintegrability and delocalization in open quantum systems with symmetry.asymmetric hopping . . . .
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.
Among the few lessons that Physics teaches us on a daily basis, one in particular is hard to miss: thermalization processes are ubiquitous. With the same degree of certainty, one can predict an apple to fall onto the ground if its stalk is cut, or a cup of tea to be found cold if left on the table for too long. In this respect, generic classical and quantum mechanical systems seem to behave in the same way. This is also one of the main difficulties in the experimental observation of the bizarre effects predicted from the quantum theory, as several of these are known to disappear at finite temperatures. It has been known for a long time that thermalization is associated with “chaotic” behavior at the microscopic level, but recently systematic efforts, both theoretical and experimental, have allowed us to understand its mechanisms at an unprecedented level of accuracy. Indeed, it has been realized that thermalization generally occurs also in isolated systems, where the absence of interactions with the environment allows for first principle theoretical investigations1. In this case thermalization takes place locally, as the whole system acts as a thermal bath for its own subsystems. Theoretical research has been motivated and paralleled by exciting experimental progress in cold-atom physics, which has provided robust, nearly ideal realizations of several theoretical models2. As an established piece of knowledge, recent research has now confirmed that two outstanding exceptions actually exist in the quantum realm, where thermalization is prevented to occur. These are many-body localized systems3, where disorder plays a crucial role, and integrable ones, which are protected by the existence of higher conservation laws. By their own nature, these systems display exceptional non-equilibrium features, which are not washed out by the onset of thermalization as relaxation occurs. At the core of the present thesis lie the remarkable properties of integrable systems out of equilibrium, with a special attention to physical effects which could be observed in cold-atom realizations. The work collected here is part of a large theoretical effort4 which in the past decade has focused on the study of relatively simple protocols to bring a quantum system out of equilibrium, such as the so-called quantum quench. In these “theoretical laboratories” it has been possible to provide quantitative predictions which helped us to develop an intuition on general questions regarding non-equilibrium and thermalization processes. Furthermore, in some cases these studies have highlighted interesting physical effects which could be directly probed experimentally within cold-atom settings. The contribution of the present thesis is two-fold. On the one hand, we have developed new technical tools for the study of integrable systems out of equilibrium, and for the computation of measurable physical quantities such as correlation functions. The techniques employed are mainly analytical and rooted within the mathematical structures of integrability. On the other hand, we have singled out physically relevant nonequilibrium situations where exotic, non-thermal states of matter emerge after relaxation occurs, and provided quantitative analytical predictions in these cases. A comprehensive discussion of the motivations and results of our work will be presented in Chapter 1, where we provide a complete overview of this thesis and discuss the organization of its content.
To understand in detail the relation between unitary quantum theory that describes our world at the microscopic scale and thermodynamics, which was long believed only to apply to macroscopic objects, is one of the most interesting and long-standing problems in physics. Recently, this problem has received renewed attention, in particular from the community of quantum information theory, but also from the field of statistical mechanics, inspired from stochastic thermodynamics. These results suggest that thermodynamics is also relevant for individual quantum systems, provided that they can be brought into contact with thermal baths. In this thesis, I use recently developed tools to provide new results both on fundamental questions, but also on questions which are of practical relevance for potential miniaturized thermal machines. In terms of fundamental questions, the results in this thesis contribute to understanding how the basic laws of thermodynamics and statistical mechanics can be understood directly from unitary quantum mechanics. In particular, I discuss and answer the following questions: i) How can we quantify the third law of thermodynamics using information theoretic methods? ii) How can we quantify the "thermodynamic value" of a state-transition in quantum systems? iii) How can we axiomatically characterize the non-equilibrium free energy and relative entropy? iv) How can we justify statistical ensembles from an operational perspective, without having to introduce either some probability measures or an information theoretic entropy measure? v) How can we understand the equilibration of closed quantum systems, how long does it take and how difficult is it to avoid? In the second part of the thesis I discuss in detail how experimental restrictions, which become important at the quantum scale, influence the ultimate thermodynamic bounds for thermal machines. In particular, the results in this thesis provide thermodynamic bounds on work-extraction and efficiencies of thermal machines in situations where 1.) an experimenter only has limited field strengths available, 2.) an experimenter cannot control the interactions between particles, but external fields arbitrarily well, and 3.) situations in which a small quantum system can only be strongly coupled to heat baths. These bounds are tight and I provide explicit examples illustrating the different behaviours. Finally, I come back to a classic problem in statistical physics: The emergence of spontaneous symmetry breaking. Here, I provide general and rigorous new results that show how symmetry-breaking stationary states emerge from fluctuations in order parameters in dissipative lattice models.
Deriving conditions under which a macroscopic system thermalizes directly from the underlying quantum many-body dynamics of its microscopic constituents is a long-standing challenge in theoretical physics. The well-known eigenstate thermalization hypothesis (ETH) is presumed to be a key mechanism, but has defied rigorous verification for generic systems thus far. A weaker variant (weak ETH), by contrast, is provably true for a large variety of systems, including even many integrable models, but its implications with respect to the problem of thermalization are still largely unexplored. Here we analytically demonstrate that systems satisfying the weak ETH exhibit thermalization for two very natural classes of far-from-equilibrium initial conditions: the overwhelming majority of all pure states with a preset nonequilibrium expectation value of some given local observable, and the Gibbs states of a Hamiltonian which subsequently is subject to a quantum quench in the form of a sudden change of some local system properties.