We review selected advances in the theoretical understanding of complex quantum many-body systems with regard to emergent notions of quantum statistical mechanics. We cover topics such as equilibration and thermalisation in pure state statistical mechanics, the eigenstate thermalisation hypothesis, the equivalence of ensembles, non-equilibration dynamics following global and local quenches as well as ramps. We also address initial state independence, absence of thermalisation, and many-body localisation. We elucidate the role played by key concepts for these phenomena, such as Lieb-Robinson bounds, entanglement growth, typicality arguments, quantum maximum entropy principles and the generalised Gibbs ensembles, and quantum (non-)integrability. We put emphasis on rigorous approaches and present the most important results in a unified language.
We review selected advances in the theoretical understanding of complex quantum many-body systems with regard to emergent notions of quantum statistical mechanics. We cover topics such as equilibration and thermalisation in pure state statistical mechanics, the eigenstate thermalisation hypothesis, the equivalence of ensembles, non-equilibration dynamics following global and local quenches as well as ramps. We also address initial state independence, absence of thermalisation, and many-body localisation. We elucidate the role played by key concepts for these phenomena, such as Lieb-Robinson bounds, entanglement growth, typicality arguments, quantum maximum entropy principles and the generalised Gibbs ensembles, and quantum (non-)integrability. We put emphasis on rigorous approaches and present the most important results in a unified language.
This thesis fathoms out the capabilities of the theory of quantum mechanics to explain thermodynamic behavior. It covers in particular equilibration and thermalization in closed quantum systems, typicality, time scales for equilibration, quantum integrability and its connection to thermalization, decoherence, and a maximum entropy principle. Together, the presented results form the body of the theory of pure state quantum statistical mechanics. With almost 300 references, ranging from the groundbreaking works of the early 20th century to the most recent discoveries (up to 2013), this work arguably constitutes the most comprehensive review of the literature on equilibration and thermalization in closed quantum systems. All results are presented in a unified notation and many are slightly strengthened or generalized.
During recent years the interest to dynamics of quantum systems has grown considerably. Quantum many body systems out of equilibrium often manifest behavior, different from the one predicted by standard statistical mechanics and thermodynamics in equilibrium. Since the dynamics of a many body quantum system typically involve many excited eigenstates, with a non-thermal distribution, the time evolution of such a system provides an unique way for investigation of non-equilibrium quantum statistical mechanics. Last decade such new subjects like quantum quenches, thermalization, pre-thermalization, equilibration, generalized Gibbs ensemble, etc. are among the most attractive topics of investigation in modern quantum physics. One of the most interesting themes in the study of dynamics of quantum many-body systems out of equilibrium is connected with the recently proposed important concept of dynamical quantum phase transitions. During the last few years a great progress has been achieved in studying of those singularities in the time dependence of characteristics of quantum mechanical systems, in particular, in understanding how the quantum critical points of equilibrium thermodynamics affect their dynamical properties. Dynamical quantum phase transitions reveal universality, scaling, connection to the topology, and many other interesting features. Here we review the recent achievements of this quickly developing part of low temperature quantum physics. The study of dynamical quantum phase transitions is especially important in context of their connection to the problem of the modern theory of quantum information, where namely non-equilibrium dynamics of many-body quantum system plays the major role.
During recent years the interest to dynamics of quantum systems has grown considerably. Quantum many body systems out of equilibrium often manifest behavior, different from the one predicted by standard statistical mechanics and thermodynamics in equilibrium. Since the dynamics of a many-body quantum system typically involve many excited eigenstates, with a non-thermal distribution, the time evolution of such a system provides an unique way for investigation of non-equilibrium quantum statistical mechanics. Last decade such new subjects like quantum quenches, thermalization, pre-thermalization, equilibration, generalized Gibbs ensemble, etc. are among the most attractive topics of investigation in modern quantum physics. One of the most interesting themes in the study of dynamics of quantum many-body systems out of equilibrium is connected with the recently proposed important concept of dynamical quantum phase transitions. During the last few years a great progress has been achieved in studying of those singularities in the time dependence of characteristics of quantum mechanical systems, in particular, in understanding how the quantum critical points of equilibrium thermodynamics affect their dynamical properties. Dynamical quantum phase transitions reveal universality, scaling, connection to the topology, and many other interesting features. Here we review the recent achievements of this quickly developing part of low-temperature quantum physics. The study of dynamical quantum phase transitions is especially important in context of their connection to the problem of the modern theory of quantum information, where namely non-equilibrium dynamics of many-body quantum system plays the major role.
The ability of classical thermodynamics to predict phenomena using only handful of parameters is miraculous. Moreover, this framework contrasts with other theories with its ability to describe our day to day experience, i. e. we know intuitively that an ice cube melts in a cold beverage on a hot summer day. A similarly powerful yet significantly less intuitive theory is quantum mechanics which describes the world at its tiniest length scales. Given this foundational position, we would expect to be able to derive all physical theories from quantum mechanics only. Unfortunately this would require knowing the quantum state of all particles which is costly as quantum mechanics requires resources growing exponentially in the system size to describe even relatively innocent many-body systems. Given this uncontrollable growth of parameters, it is highly desirable to apply the principles of thermodynamics to quantum many-body systems as a means to describe them ef- ficiently. Yet, the two theories are not obviously compatible. Subtle differences such as the definition of thermalisation already puzzled founding fathers like John von Neumann. While these have been understood by now, the focus shifted to gaining a thorough understanding of the process of thermalisation in the quantum regime, meaning that the state of the system can be described by a thermal ensemble locally after an initial out-of-equilibrium situation. In the recent past the field of quantum thermodynamics and specifically out-of-equilibrium dynamics has witnessed many successes notably the advent of highly controllable analogue quantum simulators which allow to probe out-of-equilibrium physics to an unprecedented precision. It has been established that thermalisation takes place in quantum many-body systems, but the underlying reasons and a comprehensive theory are still elusive. Only very recently, a class of systems exhibiting many-body localisation has been identi ed that defy this paradigm of thermalisation by localising their constituent particles despite many-body interactions. This can be captured using an effective description in terms of local constants of motion. Understanding the emergence of these physically relevant operators that hinder thermalisation hopefully sheds light also on the mechanism of thermalisation itself. In this thesis, we investigate the relation between thermodynamics and quantum mechanics in the context of out-of-equilibrium physics for many-body localisation. We give numerical evidence for the stability of localisation in large, closed onedimensional chains. Furthermore, we connect the established phenomenology of many-body localisation to experiments with ultra-cold atoms in optical lattices and give a blue print of measurements that can be performed with existing technology to show that the simulator indeed exhibits many-body localisation. We then set out to obtain a deeper understanding of the underlying theory of many-body localisation by devising an algorithm that explicitly constructs exact constants of motion allowing us to put the existing theoretical framework of many-body localisation to the test. It is our firm belief that understanding local constants of motion is indispensable to not only understand thermalisation in quantum many-body systems but is also of great use in other fields. A connection of this type is displayed by an extension of our algorithm to finding edge modes in spin chains with symmetry protected order.
One of the main questions of research on quantum many-body systems following unitary out of equilibrium dynamics is to find out how local expectation values equilibrate in time. For non-interacting models, this question is rather well understood. However, the best known bounds for general quantum systems are vastly crude, scaling unfavorable with the system size. Nevertheless, empirical and numerical evidence suggests that for generic interacting many-body systems, generic local observables, and sufficiently well-behaved states, the equilibration time does not depend strongly on the system size, but only the precision with which this occurs does. In this discussion paper, we aim at giving very simple and plausible arguments for why this happens. While our discussion does not yield rigorous results about equilibration time scales, we believe that it helps to clarify the essential underlying mechanisms, the intuition and important figure of merits behind equilibration. We then connect our arguments to common assumptions and numerical results in the field of equilibration and thermalization of closed quantum systems, such as the eigenstate thermalization hypothesis as well as rigorous results on interacting quantum many-body systems. Finally, we complement our discussions with numerical results - both in the case of examples and counter-examples of equilibrating systems.
Recent years have seen an enormously revived interest in the study of thermodynamic notions in the quantum regime. This applies both to the study of notions of work extraction in thermal machines in the quantum regime, as well as to questions of equilibration and thermalisation of interacting quantum many-body systems as such. In this work we bring together these two lines of research by studying work extraction in a closed system that undergoes a sequence of quenches and equilibration steps concomitant with free evolutions. In this way, we incorporate an important insight from the study of the dynamics of quantum many body systems: the evolution of closed systems is expected to be well described, for relevant observables and most times, by a suitable equilibrium state. We will consider three kinds of equilibration, namely to (i) the time averaged state, (ii) the Gibbs ensemble and (iii) the generalised Gibbs ensemble, reflecting further constants of motion in integrable models. For each effective description, we investigate notions of entropy production, the validity of the minimal work principle and properties of optimal work extraction protocols. While we keep the discussion general, much room is dedicated to the discussion of paradigmatic non-interacting fermionic quantum many-body systems, for which we identify significant differences with respect to the role of the minimal work principle. Our work not only has implications for experiments with cold atoms, but also can be viewed as suggesting a mindset for quantum thermodynamics where the role of the external heat baths is instead played by the system itself, with its internal degrees of freedom bringing coarse-grained observables to equilibrium.