The phenomenon of universality is one of the most striking in many-body physics. Despite having sometimes wildly different microscopic constituents, systems can nonetheless behave in precisely the same way, with only the variable names interchanged. The canonical examples are those of liquid boiling into vapor and quantum spins aligning into a ferromagnet; despite their obvious differences, they nonetheless both obey quantitatively the same scaling laws, and are thus in the same universality class. Remarkable though this is, universality is generally a phenomenon limited to thermodynamic equilibrium, most commonly present at transitions between different equilibrium phases. Once out of equilibrium, the fate of universality is much less clear. Can strongly non-equilibrium systems behave universally, and are their universality classes different from those familiar from equilibrium? How is quantum mechanics important? This dissertation attempts to address these questions, at least in a small way, by showing and analyzing universal phenomena in several classes of non-equilibrium quantum systems.
The phenomenon of universality is one of the most striking in many-body physics. Despite having sometimes wildly different microscopic constituents, systems can nonetheless behave in precisely the same way, with only the variable names interchanged. The canonical examples are those of liquid boiling into vapor and quantum spins aligning into a ferromagnet; despite their obvious differences, they nonetheless both obey quantitatively the same scaling laws, and are thus in the same universality class. Remarkable though this is, universality is generally a phenomenon limited to thermodynamic equilibrium, most commonly present at transitions between different equilibrium phases. Once out of equilibrium, the fate of universality is much less clear. Can strongly non-equilibrium systems behave universally, and are their universality classes different from those familiar from equilibrium? How is quantum mechanics important? This dissertation attempts to address these questions, at least in a small way, by showing and analyzing universal phenomena in several classes of non-equilibrium quantum systems.
Universality and scaling laws are hallmarks of equilibrium phase transitions and critical phenomena. However, extending these concepts to non-equilibrium systems is an outstanding challenge. Despite recent progress in the study of dynamical phases, the universality classes and scaling laws for non-equilibrium phenomena are far less understood than those in equilibrium. In this work, using a trapped-ion quantum simulator with single-spin resolution, we investigate the non-equilibrium nature of critical fluctuations following a quantum quench to the critical point. We probe the scaling of spin fluctuations after a series of quenches to the critical Hamiltonian of a long-range Ising model. With systems of up to 50 spins, we show that the amplitude and timescale of the post-quench fluctuations scale with system size with distinct universal critical exponents, depending on the quench protocol. While a generic quench can lead to thermal critical behavior, we find that a second quench from one critical state to another (i.e. a double quench) results in a new universal non-equilibrium behavior, identified by a set of critical exponents distinct from their equilibrium counterparts. Our results demonstrate the ability of quantum simulators to explore universal scaling beyond equilibrium.
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.
Universality is one of the central organising principles of modern physics, explaining why systems with vastly different microscopic constituents can exhibit identical large-scale behaviour. While the classification of equilibrium critical phenomena through hydrodynamics and the renormalization group (RG) is now well established, our understanding of universality far from equilibrium remains far less developed. In recent years, however, rapid progress - driven in large part by developments in active and living matter - has uncovered a growing range of genuinely nonequilibrium universality classes (UCs) with no equilibrium counterparts. In this review, we present a pedagogical and unified introduction to hydrodynamic and RG approaches to nonequilibrium many-body systems. We first show how hydrodynamic theories can be systematically constructed from symmetry and conservation laws alone. We then introduce perturbative dynamic RG methods and demonstrate how hydrodynamic theories are organised into distinct UCs according to their scaling behaviour. Building on these foundations, we review the diverse nonequilibrium UCs uncovered since 2015, while emphasizing the conceptual connections and unifying physical principles underlying their emergence. We conclude by discussing open theoretical and experimental challenges for the field.
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.
The circumstances under which a system reaches thermal equilibrium, and how to derive this from basic dynamical laws, has been a major question from the very beginning of thermodynamics and statistical mechanics. Despite considerable progress, it remains an open problem. Motivated by this issue, we address the more general question of equilibration. We prove, with virtually full generality, that reaching equilibrium is a universal property of quantum systems: almost any subsystem in interaction with a large enough bath will reach an equilibrium state and remain close to it for almost all times. We also prove several general results about other aspects of thermalization besides equilibration, for example, that the equilibrium state does not depend on the detailed microstate of the bath.