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 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.
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.
Technological and scientific advances have given rise to an era in which coherent quantum-mechanical phenomena can be probed and experimentally-realised over unprecedented timescales in condensed matter physics. In turn, scientific interest in non-equilibrium dynamics and irreversibility signatures of thermodynamics has taken place in recent decades, particularly in relation to cold-atom platforms and thermoelectric devices. In this PhD thesis I summarise some of the most important results obtained over the duration of my PhD, on the topic of thermodynamics involving interacting many-body quantum systems. The topics of discussion encompass three main themes: spin/particle transport in non-integrable systems, explorations in eigenstate thermalisation and finite-temperature transport in autonomous thermal machines. By questioning the effect of local integrability-breaking perturbations, I describe the subtle effects that may rise from conservation laws and their connection to linear response transport using microscopic models. Then, I describe thermalisation in the context of the eigenstate thermalisation hypothesis and its consequence to multipartite entanglement and high-order correlation functions. Finally, motivated by the necessity of describing quantum thermal machines in the finite-temperature regime, I introduce a novel tensor-network based method to investigate thermodynamic properties of autonomous machines that employ interacting many-body systems as working media.
Quantum systems exhibit unique behavior, often defying classical expectations. Their dynamics are profoundly shaped by the interplay of quantum and thermal fluctuations, making precise predictions challenging. This paper explores the non-equilibrium thermodynamics of quantum systems, focusing on the development of a novel framework to address this challenge. We propose a 'quantum-thermal coupling' algorithm, designed to capture the complex dynamics of these systems, offering a paradigm shift in our understanding of quantum phenomena in a dynamically changing environment. The core claim is to provide a robust and adaptable approach for modeling and predicting the behavior of quantum systems in non-equilibrium conditions, considering the intricate interplay of quantum and thermal effects. This work builds upon existing research in quantum mechanics and thermodynamics, addressing limitations in current models and establishing a foundational step towards a more complete understanding of these systems.
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.
Recent advancements in atomic, molecular, and optical platforms have propelled the study of many-body physics into a new era. The ability to construct large ensembles of precisely manipulated elementary components is revolutionizing both the toolbox and the mindset for exploring the intricate relationship between microscopic details and macroscopic phenomena. This exploration lies at the heart of many-body physics, particularly in the context of out-of-equilibrium many-body dynamics. Indeed, the past decades have witnessed a resurgence of classic topics alongside the emergence of new inquiries in the field of many-body dynamics, covering a spectrum from fundamental dynamical properties to their practical applications in quantum technology. This thesis contributes to such a vibrant field by employing a combination of analytical and numerical methodologies to investigate various aspects of out-of-equilibrium dynamics, such as thermalization, hydrodynamics, and responses to drives. Furthermore, this study is intimately intertwined with experimental investigation, where empirical observations enrich theoretical frameworks, and theoretical predictions guide experimental pursuits and practical applications.