The thermal properties of ice, liquid water and steam are at odds with statistical theories applied to many-body systems. Here, these properties are quantitatively explained with a bulk-scale matter field emerging from the indefinite status of the microscopic constituents. Such a field is characterized by its symmetry in spacetime, its degree of degeneracy and its eigenstates. There are several one-to-one correspondences bridging outcomes of classical and quantum measurements. (i) The heat capacities are linked to the symmetry of the field for each phase of water. (ii) The latent heats are linked to the change of the degree of degeneracy for each transition. (iii) The critical temperatures are linked to the eigenstates of the potential operator. The matter field leads to a complete representation of the phases of water, free of hidden parameters and statistical ignorance
The thermal properties of ice, liquid water and steam are at odds with statistical theories applied to many-body systems. Here, these properties are quantitatively explained with a bulk-scale matter field emerging from the indefinite status of the microscopic constituents. Such a field is characterized by its symmetry in spacetime, its degree of degeneracy and its eigenstates. There are several one-to-one correspondences bridging outcomes of classical and quantum measurements. (i) The heat capacities are linked to the symmetry of the field for each phase of water. (ii) The latent heats are linked to the change of the degree of degeneracy for each transition. (iii) The critical temperatures are linked to the eigenstates of the potential operator. The matter field leads to a complete representation of the phases of water, free of hidden parameters and statistical ignorance
Introduction We all observe phase transitions in our daily lives, with hardly a second thought. When we boil water for a cup of tea, we observe that the water is quiescent until it reaches a certain temperature (100 ℃), and then bubbles appear vigorously until all the water has turned to steam. Or after an overnight snowfall, we have watched the snow melt away when temperatures rise during the day. The more adventurous among us may have heated an iron magnet above about 760 ℃ and noted the disappearance of its magnetism. Familiar and ubiquitous as these and many related phenomena are, a little reflection shows that they are quite mysterious and not easy to understand: indeed, the outlines of a theory did not emerge until the middle of the twentieth century, and, although much has been understood since then, active research continues. Ice and water both consist of molecules of H 2 O, and we can look up all the physical parameters of a single molecule, and of the interaction between a pair of molecules, in standard reference texts. However, no detailed study of this information prepares us for the dramatic change that occurs at 0 ℃. Below 0 ℃, the H 2 O molecules of ice are arranged in a regular crystalline lattice, and each H 2 O molecule hardly strays from its own lattice site. Above 0 ℃, we obtain liquid water, in which all the molecules are moving freely throughout the liquid container at high speeds. Why do 10 23 H 2 O molecules cooperatively “decide” to become mobile at a certain temperature, leading to the phase transition from ice to water?
Introduction In this chapter we provide a comprehensive and unified treatment of the physical concepts and quantities relevant to phase transitions and critical phenomena. Some of these concepts and quantities have already been discussed in the particular contexts of magnetism (Chapter 7), or liquid crystals (Chapter 9, We conclude the chapter with a brief reference to the application of renormalisation group ideas to critical phenomena. The resulting reconciliation of the predictions of classical theories with experiments and exact or quasi-exact solutions of model systems is a major triumph of twentieth-century physics. Phases and phase transitions Thermodynamics habitually deals with equilibrium systems that are uniform on the macroscopic scale – i.e., where the structure, density and composition are the same throughout the bulk of the system, which is many times greater than a single atom or molecule. Because the system is in equilibrium, intensive thermodynamic variables such as the temperature or chemical potential are also spatially uniform. Such a system is said to consist of a single phase – it is a one-phase system. Phases are what used to be called the states of matter. Solids, liquids, gases and plasmas are four commonly occurring phases – they correspond to the four elements of Aristotelian physics: earth, water, air and fire, respectively. There are, however, many other, less easily identified phases, some of which we have already encountered in this book, such as the ferromagnetic, antiferromagnetic, nematic or superconducting phases. On varying the temperature, pressure and chemical potential(s) of a system, properties such as its structure, density, composition, compressibility, specific heat, etc., in general change smoothly. Yet under certain conditions the changes may be abrupt – some of these properties may exhibit discontinuities, or even divergences, at a phase transition . Phase transitions are among the most spectacular phenomena of nature. Consider, for instance, the phase sequence of water on heating: from ice, an opaque, slippery, rock-like solid; to colourless, incompressible, liquid water, which nevertheless easily flows under gravity; to water vapour, an invisible and very low-density (greenhouse!) gas, hence of ill-defined volume.
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.
Classical and quantum criticality In most cases the phases of quantum field theories, in particular those of interest in condensed matter physics, can be described in terms of the behavior of local observables, such as order parameters or currents that transform properly under the symmetries of the theory. Quantum and thermal phase transitions are characterized by the behavior of these observables as a function of temperature and of the coupling constants of the theory. The phase transitions themselves, quantum or thermal, are classified into universality classes, which are represented by the critical exponents which specify the scaling laws of the expectation values of the observables. Historically, the development of this approach to critical behavior goes back to the Landau theory of critical behavior. It acquired its most complete form with the development of the renormalization group (RG) in the late 1960s and early 1970s. It is the centerpiece of Wilson’s approach to quantum field theory, in which all local quantum field theories are defined by the scaling regime of a physical system near a continuous phase transition. From this point of view there is no fundamental difference between classical (or thermal) phase transitions, which are described by the theory of classical critical behavior, and quantum phase transitions. For example, the expectation value of a local order parameter M as the thermal phase transition is approached from below behaves as M ~ (Tc − T) β . Here T c is the critical temperature and β is a critical exponent that depends on the universality class of the thermal phase transition and on the dimensionality of space. While quantum mechanics can play a key role in the existence of the ordered phase, e.g. superfluidity and superconductivity are macroscopic manifestations of essentially quantum-mechanical phenomena, the thermal transition itself is governed entirely by classical statistical mechanics, and quantum mechanics plays a role in setting the value of non-universal quantities such as the critical temperature, etc. On the other hand, in the case of a quantum phase transition, the order parameter M has a similar scaling behavior as a function of the coupling constant, M~(gc−g)β˜, where g is the coupling constant, g c is the critical coupling constant, and β˜ is a critical exponent that depends on the universality class of the quantum phase transition. Here we assume that M has a non-vanishing expectation value only for g < g c .
Aim and Scope of this Book A plethora of systems exhibit phase transitions as the temperature or some other parameter is changed. Examples range from the ice-water phase transition observed in our daily life to the loss of ferromagnetism in iron or to the more sophisticated Mott insulator-superfluid phase transition observed in optical lattices [343]. The last five decades have witnessed a tremendous upsurge in the studies of phase transitions at finite temperature [727, 149, 333, 136, 494, 541, 556]. The success of Landau-Ginzburg theories and the concepts of spontaneous symmetry breaking and the renormalization group [27, 410, 821, 578] in explaining many of the finite temperature phase transitions occurring in nature has been spectacular. In this book, we will consider only a subclass of phase transitions called quantum phase transitions (QPTs) [154, 658, 725, 799, 185, 63, 62, 66, 141, 744] and we will discuss these mainly from the view point of recent studies of information and dynamics. QPTs are zero temperature phase transitions which are driven by quantum fluctuations and are usually associated with a non-analyticity in the ground state energy density of a quantum many-body Hamiltonian. We will focus on continuous QPTs where the order parameter vanishes continuously at the quantum critical point (QCP) at some value of the parameters which characterize the Hamiltonian. We will not discuss first order quantum phase transitions associated with an abrupt change in the order parameter. Usually, a first order phase transition is characterized by a finite discontinuity in the first derivative of the ground state energy density. A continuous QPT is similarly characterized by a finite discontinuity, or divergence, in the second derivative of the ground state energy density, assuming that the first derivative is continuous. This is of course the classical definition; we will later mention some QPTs where the ground state energy density is not necessarily singular.
The thermal or equilibrium ensemble is one of the most ubiquitous states of matter.For models comprised of many locally interacting quantum particles, it describes a wide range of physical situations, relevant to condensed-matter physics, high-energy physics, quantum chemistry and quantum computing, among others.We give a pedagogical overview of some of the most important universal features about the physics and complexity of these states, which have the fact that interactions in the Hamiltonian are short ranged at its core.We focus on mathematically rigorous statements, many of them inspired by ideas and tools from quantum information theory.These include bounds on their correlations, the form of the subsystems, various statistical properties, and the performance of classical and quantum algorithms.We also include a summary of a few of the most important technical tools, as well as some self-contained proofs.