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This chapter discusses ferroic materials, which are the first of the three types of nonlinear-response materials discussed in this book, the other two being soft materials and nanostructured materials. Their nonlinear properties are a consequence of one or more ferroic phase transitions that can occur in them. The chapter begins with an introduction to phase transitions and critical phenomena in crystals. Ferroic phase transitions are those that involve a change of point-group symmetry in a nondisruptive manner. Their thermodynamic classification is described, which helps divide ferroic materials into ferroelectrics, ferromagnetics, ferroelastics, ferrobielastics, ferroelastoelectrics, etc. The poling process for ferroic ceramics is explained. The domain structure of ferroics, and the possibilities of tailoring this structure to advantage are discussed. The importance of multiferroics in smart-structures research is emphasized. There are also sections on spin-glasses, shape-memory alloys and ceramics, and relaxor ferroelectrics. The book has helpful appendices on crystallographic symmetry and on tensor properties.

Thermodynamic measurements for a limited sample of crystalline mixed-stack 1:1 π-molecular compounds show that most are enthalpy-stabilized, some entropy-stabilized, and a few both enthalpy and entropy-stabilized. Correlation of these thermodynamic results with crystal structures remains a task for the future. Combination of optical spectroscopic methods at very low temperatures and electron spin resonance measurements have provided proof of Mulliken’s theory also for the solid state. The room temperature crystal structures of 1:1 π-molecular compounds are not necessarily representative of the entire range of pressure-temperature behaviour of these materials. There are often hints of disorder (usually of the donor) in the room temperature structures, and these have been correlated for a few systems with disorder-to-order transitions (thermodynamically second-order, following Ehrenfest) that occur on cooling. These have been studied by a combination of calorimetric, diffraction, and resonance techniques. Despite overall similarities, each system surveyed has its own individual characteristics. A number of 1:1 π-molecular compounds have been shown to transform to quasi-plastic phases on heating. A small number of mixed stack 1:1 π-molecular compounds with neutral ground states have been shown to transform to ionic structures on cooling or application of pressure.

This chapter presents a self-contained account of the slicing method that allows the exhibition of a lower bound for high-dimensional problems through their one-dimensional sections. After computing a family of one-dimensional limit problems, an optimization is performed through an argument characterizing the supremum of a family of measures. The upper inequality is obtained by a density argument whenever recovery sequences have a one-dimensional form. This method can be applied to the high-dimensional gradient theory of phase transitions.

A very important group of ferroelectrics is that known as the perovskites from the mineral perovskite CaTiO₃ (which itself is actually a distorted perovskite structure). The perfect perovskite structure is an extremely simple one with general formula ABO₃, where A is a monovalent or divalent metal and B is a tetra- or pentavalent one. The off-centring of the B-cation is a relatively independent process that can occur in any structure built from the hard octahedra. It is this off-centering which leads to the presence of dipoles and to both ferroelectricity and antiferroelectricity. This chapter looks at the oxygen octahedron and the ferroelectric perovskites, including barium titanate, potassium niobate, potassium tantalate niobate, lead titanate, sodium niobate, potassium tantalate, and strontium titanate. Antiferroelectric and cell-doubling perovskites such as lead zirconate and lithium tantalate are also discussed, along with zone-boundary lattice-mode condensation, local-mode analysis, low-temperature polarization, stoichiometry, tungsten-bronze-type structures, and diffuse phase transitions.

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.

This chapter discusses phase diagrams and phase transitions in lyotropic liquid crystals from the point of view of symmetry transformations between periodically ordered structures. In some systems, amphiphilic aggregates may suffer a drastic reorganization, with modifications of geometry, and without a group-subgroup relation between the newly formed and the initial structures. There are some examples of these reconstructive phase transitions involving lamellar, hexagonal, cubic, and sponge phases. In other cases, there are equilibrium group-subgroup relations between different structures, as in the weak first-order nematic-isotropic transitions, which are quite well described by the Landau expansion. On the basis of proposals by Mettout, Toledano, and co-workers, the chapter presents a discussion of an extension of the standard Landau approach, which is adequate to account for the reconstructive phase transitions. In particular, there are examples of lamellar-tetragonal, lamellar-cubic, and lamellar-hexagonal transitions, and discussions of direct and reversed mesophases.

This chapter focuses on phase transitions, discussing latent heat and deriving the Clausius-Clapeyron equation. It discusses the criteria for stability and metastability, and derives the Gibbs phase rule. It introduces colligative properties and classifies the different types of phase transition.

Near the temperature of a second-order structural phase transition, and in particular near the Curie point of a ferroelectric, anomalously strong ultrasonic absorption occurs. This chapter considers the perturbation produced on an acoustic travelling wave by those interactions with a soft optic (ferroelectric) lattice variable. The resulting anomalies affect both the real and imaginary parts of the acoustic frequency and are observable respectively as shifts in acoustic velocity (or equivalently elastic moduli) and in ultrasonic attenuation. In general, the existence of coupling terms, whether in low or high order, leads to characteristic anomalies in other observables which are essentially driven by the primary soft mode. Acoustic anomalies near ferroelectric phase transitions are discussed, including anomalies in sound velocity and elastic constants, along with extrinsic (improper) ferroelectrics and ferroelastics, gadolinium molybdate and its isomorphs, and classification of ferro-materials.

Phase transitions with sharp interfaces are interpreted as segmentation problems, and are approximated with perturbations of non-convex energies (gradient theory of phase transitions). The Modica-Mortola theorem is proved, with different formulas characterizing the segmentation energy (optimal-profile problems).

This chapter discusses the quantum phase transition using the example of superconductor-insulator transition. For the 3D case, a version of the Ginzburg-Landau formalism is formulated from both normal and anomalous diffusion. The description of such transition in the case of 2D superconductors is very specific and strongly differs from the 3D case. The ideas of boson and fermion mechanisms of the T_c suppression in 2D cases are presented, and their predictions are compared with the experimental conditions.

This chapter starts off with a discussion of the specifics of superconductivity in ultrasmall superconducting grains. The method of optimal fluctuations in the vicinity of T_c is then introduced, and applied to the study of the formation of superconducting drops in a system with quenched disorder or in strong magnetic fields. The exponential DOS tail in a superconductor with quenched disorder is calculated. Properties of Josephson coupled superconducting grains and drops are discussed. The XY-model for granular superconductor and the GL description of the granular superconductor are formulated. The broadening of superconducting transition by the quenched disorder is found. The final part of the chapter focuses on the specifics of the quantum phase transition in granular superconductors. It discusses Coulomb suppression of superconductivity in the array of tunnel coupled granules, properties of superconducting grains in the normal metal matrix, and phase transition in disordered superconducting film in strong magnetic field.

This chapter argues that the widespread notion that the discipline of condensed matter physics is devoted to deriving the properties of complex many-body systems from that of their atomic-level components is a myth, and that the analogy of map-making is much more appropriate. After a brief discussion of the experimental techniques available in this area and of the properties of the simplest condensed-matter systems (gases, simple liquids, and crystalline solids), attention is focussed first on the problem of second- and first-order phase transitions, and secondly on issues arising in amorphous (glassy) materials and their possible relevance to biological systems. The chapter concludes by introducing the phenomena of superconductivity and superfluidity, and discussing the general problem of interfacing condensed-matter systems with their environments.

As an introduction to the physics of phase transitions and critical phenomena, this chapter explains a number of basic and fundamental ideas such as phases, phase transitions, phase diagrams, universality, and critical phenomena. Especially important is the concept of order parameter, a quantity that measures the degree of asymmetry in the broken symmetry phase. Intuitive accounts are given to the concepts of coarse-graining, and scale and renormalization group transformations, which are powerful, systematic tools to analyze critical behaviour of macroscopic systems. Also explained are several spin and lattice gas model systems, on the basis of which phase transitions and critical phenomena will be studied.

This chapter finally deals with the concept of spin glasses. The intention is not to provide anything approaching a thorough history of the subject. The field today is broad, with threads and subthreads extending in a multitude of different directions. Rather, the chapter focuses on a relatively narrow part of the overall subject. It discusses some of the history of their discovery, their basic properties and experimental phenomenology, and some of the mysteries surrounding them. It introduces some of the basic theoretical constructs that underlie much of the discussion in later chapters. Topics covered include dilute magnetic alloys and the Kondo effect, nonequilibrium and dynamical behavior, mechanisms underlying spin glass behavior, the Edwards–Anderson Hamiltonian, frustration, dimensionality and phase transitions, broken symmetry and the Edwards–Anderson Order Parameter, and energy landscapes and metastability.