Quantum mechanics provides various pictures for understanding physical phenomena, yet the full potential of these pictures' equivalence remains underutilized. This work introduces a novel picture, opposite to the interaction picture, achieved through a transformation of the time evolution operator. This new approach permits the manipulation of the relative weight of Hamiltonian components. Utilizing picture equivalence, we investigate dynamical quantum phase transitions and quantum thermodynamics. Our findings reveal a significant relationship between dynamical topological quantum phase transitions and accidental ones through time-reversal operations, offering new insights into their fundamental connections. Moreover, our picture-based transformation addresses the challenge of defining thermodynamic quantities in systems with strong system-reservoir coupling, maintaining the robustness of weak coupling analyses.
Quantum mechanics provides various pictures for understanding physical phenomena, yet the full potential of these pictures' equivalence remains underutilized. This work introduces a novel picture, opposite to the interaction picture, achieved through a transformation of the time evolution operator. This new approach permits the manipulation of the relative weight of Hamiltonian components. Utilizing picture equivalence, we investigate dynamical quantum phase transitions and quantum thermodynamics. Our findings reveal a significant relationship between dynamical topological quantum phase transitions and accidental ones through time-reversal operations, offering new insights into their fundamental connections. Moreover, our picture-based transformation addresses the challenge of defining thermodynamic quantities in systems with strong system-reservoir coupling, maintaining the robustness of weak coupling analyses.
Abstract: This study proposes a unified framework that integrates topological quantum field theories with statistical mechanics to provide a comprehensive understanding of quantum states and dynamics. The problem of reconciling the quantum mechanical description of microscopic systems with macroscopic statistical behavior is addressed by exploring the relationship between topological invariants and thermodynamic properties in quantum systems. The methodology involves the development of a mathematical model that combines the abstract principles of topological quantum field theory with the probabilistic approaches of statistical mechanics, utilizing tensor calculus and advanced mathematical physics tools. Key results indicate that the proposed framework offers new insights into quantum phase transitions, critical phenomena, and the thermodynamic behavior of quantum systems, especially in systems exhibiting topological phases. The framework is further evaluated through a series of theoretical simulations, demonstrating its applicability to quantum information science, condensed matter physics, and quantum technology applications. The implications of this research are significant, providing a foundation for the development of new quantum technologies, such as quantum computing and quantum materials, and contributing to a deeper understanding of the connections between topological properties and statistical mechanics in quantum systems. Keywords: Topological Quantum Field Theory, Statistical Mechanics, Quantum States, Quantum Dynamics, Phase Transitions, Critical Phenomena, Quantum Computing, Quantum Materials, Thermodynamics, Topological Phases, Quantum Information Science.
Identifying or constructing a fine-grained microscopic theory that will emerge under specific conditions to a known macroscopic theory is always a formidable challenge. Thermodynamics is perhaps one of the most powerful theories and best understood examples of emergence in physical sciences, which can be used for understanding the characteristics and mechanisms of emergent processes, both in terms of emergent structures and the emergent laws governing the effective or collective variables. Viewing quantum mechanics as an emergent theory requires a better understanding of all this. In this work we aim at a very modest goal, not quantum mechanics as thermodynamics, not yet, but the thermodynamics of quantum systems, or quantum thermodynamics. We will show why even with this minimal demand, there are many new issues which need be addressed and new rules formulated. The thermodynamics of small quantum many-body systems strongly coupled to a heat bath at low temperatures with non-Markovian behavior contains elements, such as quantum coherence, correlations, entanglement and fluctuations, that are not well recognized in traditional thermodynamics, built on large systems vanishingly weakly coupled to a non-dynamical reservoir. For quantum thermodynamics at strong coupling, one needs to reexamine the meaning of the thermodynamic functions, the viability of the thermodynamic relations and the validity of the thermodynamic laws anew. After a brief motivation, this paper starts with a short overview of the quantum formulation based on Gelin & Thoss and Seifert. We then provide a quantum formulation of Jarzynski's two representations. We show how to construct the operator thermodynamic potentials, the expectation values of which provide the familiar thermodynamic variables. Constructing the operator thermodynamic functions and verifying or modifying their relations is a necessary first step in the establishment of a viable thermodynamics theory for quantum systems. We mention noteworthy subtleties for quantum thermodynamics at strong coupling, such as in issues related to energy and entropy, and possible ambiguities of their operator forms. We end by indicating some fruitful pathways for further developments.
Quantum Complexity Theory (QCT) aims to provide a framework for understanding and predicting the behavior of complex quantum systems. This paper explores a novel approach based on topological phase transitions, offering a new perspective on the study of quantum mechanics and its applications. We propose a method for modeling the system using a topological graph, analyzing transitions between states, and utilizing the inherent properties of topological phases to uncover underlying dynamics. This work integrates topology and quantum mechanics to provide a robust and insightful tool for analyzing complex quantum phenomena. The framework presented herein offers a potential pathway to developing more accurate simulations and predictions, ultimately advancing our comprehension of quantum systems.
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.
Nonequilibrium states of closed quantum many-body systems defy a thermodynamic description. As a consequence, constraints such as the principle of equal a priori probabilities in the microcanonical ensemble can be relaxed, which can lead to quantum states with novel properties of genuine nonequilibrium nature. In turn, for the theoretical description it is in general not sufficient to understand nonequilibrium dynamics on the basis of the properties of the involved Hamiltonians. Instead it becomes important to characterize time-evolution operators, which adds time as an additional scale to the problem. In this perspective article we summarize recent progress in the field of dynamical quantum phase transitions, which are phase transitions in time with temporal nonanalyticities in matrix elements of the time-evolution operator. These transitions are not driven by an external control parameter, but rather occur due to sharp internal changes generated solely by unitary real-time dynamics. We discuss the obtained insights on general properties of dynamical quantum phase transitions, their physical interpretation, potential future research directions, as well as recent experimental observations.