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In order to acquire a simple physical picture of the dynamic mechanism of a phase transition it is necessary to use the simplest of many-body approximations. It is instructive, in particular, to study the mean-field response of the model system to a time-dependent applied field. In this way, one can obtain considerable insight into the nature of the collective excitations and into the relationship between the static aspects of a phase transition and the occurrence of temperature-dependent (that is, soft) modes and of critical fluctuations. This chapter discusses the static aspects of mean-field theory and the nature of the static singularities which accompany second-order phase transitions. Mean-field dynamics are then described in terms of deviations from the equilibrium mean-field state. Correlated effective-field theory, the quasi-harmonic limit and self-consistent phonons, the deep double-well limit and the Ising model, and the pseudo-spin formalism and tunnel mode are also considered.

In this chapter a novel, rigorous approach to analyse the validity of continuum approximations for deterministic interacting particle systems is discussed. The focus is on the Boltzmann–Grad limit of ballistic annihilation, a topic which has has received considerable attention in the physics literature. In this situation, due to the deterministic nature of the evolution, it is possible that correlations build up and the mean–field approximation by the Boltzmann equation breaks down. A sharp condition on the initial distribution, which ensures the validity of the Boltzmann equation is given, together with an example demonstrating the failure of the mean-field theory if the condition is violated.

This chapter provides a review of recently-developed Dynamical Mean-Field Theory (DMFT) approaches to the general problem of strongly correlated electronic systems with disorder. The chapter first describes the standard DMFT approach, which is exact in the limit of large coordination, and explain why in its simplest form it cannot capture either Anderson localization or the glassy behavior of electrons. Various extensions of DMFT are then described, including statistical DMFT, typical medium theory, and extended DMFT, methods specifically designed to overcome the limitations of the original formulation. The chapter provides an overview of the results obtained using these approaches, including the formation of electronic Griffiths phases, the self-organized criticality of the Coulomb glass, and the two-fluid behavior near Mott-Anderson transitions. Finally, the chapter outlines research directions that may provide a route to bridge the gap between the DMFT-based theories and the complementary diffusion-mode approaches to the metal-insulator

This chapter introduces a quantum field theory for interacting boson systems. It develops a mean-field theory to study the superfluid phase. A path integral formulation is then developed to re-derive the superfuid phase, which results in a low energy effective non-linear sigma model. A renormalization group approach is introduced to study the zero temperature quantum phase transition between superfluid and Mott insulator phase, and finite temperature phase transition between superfluid and normal phase. The physics and the importance of symmetry breaking in phase transitions and in protecting gapless excitations are discussed. The phenomenon of superfluidity and superconductivity is also discussed, where the coupling to U(1) gauge field is introduced.

Methods of statistical mechanics have been enormously successful in clarifying the macroscopic properties of many-body systems. Typical examples are found in magnetic systems, which have been a test bed for a variety of techniques. This chapter introduces the Ising model of magnetic systems and explains its mean-field treatment, a very useful technique of analysis of many-body systems by statistical mechanics. Mean-field theory explained here forms the basis of the methods used repeatedly throughout this book. The arguments in the present chapter represent a general mean-field theory of phase transitions in the Ising model with uniform ferromagnetic interactions. Special features of spin glasses and related disordered systems are taken into account in subsequent chapters.

This chapter introduces mean field theory, both as a general class of models and in its specific incarnation in spin glasses, the Sherrington–Kirkpatrick model. This is undoubtedly the most theoretically studied spin glass model by far, and the best understood. For the nonphysicist the going may get a little heavy in this chapter once replica symmetry breaking is introduced, with its attendant features of many states, non-self-averaging, and ultrametricity—but an attempt is made to define and explain what all of these things mean and why replica symmetry breaking represents such a radical departure from more conventional and familiar modes of symmetry breaking. While this is a central part of the story of spin glasses proper, the nonphysicist who wants to skip the technical details can safely omit certain sections in the chapter and continue on without losing the essential thread of the discussion that follows.

This chapter discusses the problem of spin glasses. If the interactions between spins are not uniform in space, the analysis of the previous chapter does not apply. In particular, when the interactions are ferromagnetic for some bonds and antiferromagnetic for others, the spin orientation cannot be uniform in space, unlike the ferromagnetic system. Under such a circumstance it sometimes happens that spins become randomly frozen — random in space but frozen in time. This is the intuitive picture of the spin glass phase. The chapter investigates the condition for the existence of the spin glass phase as an extension of the mean-field theory. In particular, the properties of the so-called replica-symmetric solution are explained in detail for the Sherrington–Kirkpatrick (SK) model.

Thermal properties of magnets are dominated by low-lying excitations, perused systematically. Magnon spectra of elementary metals and compounds are obtained theoretically and compared with experimental data. Spin fluctuations are discussed in mean-field theory to obtain ab initio estimates of ordering temperatures for a multitude of magnetic systems. The free energy is connected with dynamic susceptibility which supplies a solid basis for the magnetic phase of ferromagnetic compounds. Methods derived to obtain Heisenberg exchange constants from first-principle calculations are compared with experimental data. Magnetic skyrmions enrich the field of magnetism and are of possible use for data technology applications. Several cases are discussed and classified showing theoretical and experimental data. For high temperatures the disordered local moment picture supplies an alternative theory for magnetism where the coherent-potential approximation is used to solve the electronic-structure problem in an alloy analogy. The basic theory is presented and discussed together with experimental data.

This chapter discusses quantum phase transitions (QPT). It starts with a brief review of thermal phase transitions, critical exponents and scaling laws. The scaling laws are then generalized to the QPT case which is also illustrated with two specific examples. The first example that of the one-dimensional Ising model in a transverse magnetic field; the second is that of the bosonic Hubbard model. Quantum Monte Carlo is described briefly and mean field theory is introduced with the help of several examples and exercises.

Phase transitions as a collective response of an ensemble, with appearance of unique stable properties spontaneously, is critical to a variety of devices: electronic, magnetic, optical, and their coupled forms. This chapter starts with a discussion of broken symmetry and its manifestation in the property changes in thermodynamic phase transition and the Landau mean-field articulation. It then follows it with an exploration of different phenomena and their use in devices. The first is ferroelectricity—spontaneous electric polarization—and its use in ferroelectric memories. Electron correlation effects are explored, and then conductivity transition from electron-electron and electron-phonon coupling and its use in novel memory and device forms. This is followed by development of an understanding of spin correlations and interactions and magnetism—spontaneous magnetic polarization. The use and manipulation of the magnetic phase transition in disk drives, magnetic and spin-torque memory as well as their stability is explored. Finally, as a fourth example, amorphous-crystalline structural transition in optical, electronic, and optoelectronic form are analyzed. This latter’s application include disk drives and resistive memories in the form of phase-change as well as those with electochemical transport.

Liquid crystal is a state of matter which has an intermediate order between liquids and crystals. While fluid in nature, the materials in liquid crystals posses an order in molecular orientation. As a result, the molecular orientation of liquid crystal is easily controlled by weak forces, a property that is extensively used in the application of liquid crystals to display devices. Liquid crystal is an example that the collective nature of soft matter is created by phase transition. This chapter discusses how the interaction between individual molecules creates spontaneous macroscopic ordering, and how it affects the material response to external forces. The phase transition in liquid crystals is an example of order–disorder transition, the general aspects of which can be seen in liquid crystals.

Bose–Einstein condensation (BEC) in cold atomic gases was first achieved experimentally in 1995. Since then there has been a surge of activity in this field, with ingenious experiments putting forth more and more astonishing results about the behaviour of matter at very cold temperatures. The theoretical investigation of BEC goes back much further, and even predates the modern formulation of quantum mechanics. It was investigated in two papers by Einstein in 1924 and 1925, respectively, following up on a work by Bose on the derivation of Planck's radiation law. Einstein's result, in its modern formulation, can be found in any textbook on quantum statistical mechanics, and was concerned with ideal, i.e., non-interacting gases. The understanding of BEC in the presence of interparticle interactions poses a formidable challenge to mathematical physics. Some progress has been made in the last ten years or so, and the purpose of this chapter is to explain part of what was achieved and how it is related to the actual experiments on cold gases.

This chapter presents two examples of the application of Boltzmann statistics to systems with nontrivial interactions between particles. The first example is a nonideal gas, treated approximately using a series expansion that we can visualize in terms of simple diagrams. The second example is a model of a ferromagnet as a collection of two-state particles interacting with their nearest neighbors. It is easy to solve this model exactly in one dimension, and to gain a semi-quantitative understanding of why the system magnetizes below a critical temperature in two or three dimensions. The most powerful tool for studying this model, however, is numerical simulation on a computer using a random-sampling algorithm based on the Boltzmann distribution.

This chapter discusses transport data on Ce- and Yb-based strongly correlated thermoelectric materials, in both the high-temperature regime, including pressure and doping dependences, and the low-temperature, Fermi liquid state. Heavy fermions and valence fluctuators at high temperature are modeled theoretically by treating the 4f-states on the Ce or Yb ions as independent scattering resonances. The low-temperature phase is described in terms of the infinite-U periodic Anderson model with SU(N) symmetry. The self-energy of the f-electrons is assumed to be local, in order to apply the formalism of dynamical mean field theory. Expanding it to linear order in frequency yields two quasiparticle branches, and the quasiparticle density of states at the Fermi level defines the Fermi liquid scale T ₀. The transport coefficients are obtained from a Sommerfeld expansion of the transport integrals and shown to follow the observed universal laws. Pressure effects in the Fermi liquid phase are discussed.

Thermodynamic scaling near the critical point is a signature of critical phenomena, and many useful applications of supercritical solvent fluids depend upon exploiting this behavior in some technologically interesting way. Near the critical point, many transport and thermodynamic properties show anomalous behavior which is usually linked to the divergence of certain thermodynamic properties, such as the fluid’s isothermal compressibility. In figures 3.1 and 3.2 we depict the near-critical behavior of both the density of xenon and the thermal conductivity of carbon dioxide, respectively, adapted from published data [1, 2]. The onset of what appear to be critical singularities in these properties is clearly evident in both instances. In this chapter, we focus upon the thermodynamic basis for this type of behavior. In the theory of critical phenomena, the limiting behavior of certain thermodynamic properties near the critical point assumes special significance. In particular, properties that diverge at the critical point are of interest, and this divergence is usually described in terms of scaling laws.