Superconductivity is a quantum state of matter that occurs through a phase transition driven by thermal fluctuations. In this state, materials show ideal electric conductivity and ideal diamagnetism to a very good approximation. Two main classes of superconductors, type I and type II, can be distinguished with regards to flux penetration under an applied magnetic field. The properties of these two types are first discussed in detail. Next, the Ginzburg–Landau theory is developed and it is shown that in the presence of a magnetic field, when the ratio of penetration and coherence lengths is smaller than 1⁄√2 the superconductor behaves as type I, while it behaves as type II when this ratio is larger than 1⁄√2. In this second case, the flux penetrates through vortices that form a hexagonal lattice. Finally, in the last part, the microscopic BCS theory is discussed in order to provide an understanding of the physical origin of superconductivity.

The chapter is an introduction to basic equilibrium aspects of phase transitions. It starts by reviewing thermodynamics and the thermodynamic description of phase transitions. Next, lattice models, such as the paradigmatic Ising model, are introduced as simple physical models that permit a mechano-statistical study of phase transitions from a more microscopic point of view. It is shown that the Ising model can quite faithfully describe many different systems after suitable interpretation of the lattice variables. Special emphasis is placed on the mean-field concept and the mean-field approximations. The deformable Ising model is then studied as an example that illustrates the interplay of different degrees of freedom. Subsequently, the Landau theory of phase transitions is introduced for continuous and first-order transitions, as well as critical and tricritical behaviour are analysed. Finally, scaling theories and the notion of universality within the framework of the renormalization group are briefly discussed.

The chapter starts by introducing the basic concepts of metastable and unstable states as well as time scales that control the occurrence of phase transitions. The limits for phase transitions taking place in equilibrium and out-of-equilibrium conditions are then established. In the latter case, thermally activated and athermal limits are distinguished associated with those situations where the transition is either driven or not driven by thermal fluctuations, respectively. Then the formal theory of the decay of metastable and unstable states in systems with conserved and non-conserved order parameters is developed. This general theory is in turn applied to the study of homogeneous and heterogeneous nucleation, spinodal decomposition and late stages of coarsening and domain growth.

The general concept of multiferroic materials as those with strong interplay between two or more ferroic properties is first introduced. Then, particular cases of materials with coupling magnetic and polar (magnetoelectric coupling), polar and structural (electrostructural coupling), and magnetic and structural (magnetostructural coupling) degrees of freedom are discussed in more detail. The physical origin of the interplay is analysed and symmetry-based considerations are used to determine the dominant coupling terms adequate to construct extended Ginzburg–Landau models that permit the determination of cross-response to multiple fields. The last part of the chapter is devoted to study morphotropic systems and morphotropic phase boundaries that separate crystallographic phases with different polar (magnetic) properties as examples of materials with electro(magneto)-structural interplay and that are expected to show giant cross-response to electric (magnetic) and mechanical fields.

Non-equilibrium phase transitions are non-thermal transitions that occur out-of-equilibrium. The chapter first discusses systems that are subjected and respond with hysteresis to an oscillating field due to a competition between driving and relaxation time scales. When the former is much shorter than the latter, a non-equilibrium transition occurs associated with the dynamical symmetry breaking due to hysteresis. A dynamical magnetic model is introduced and it is shown that the mean magnetization in a full cycle is the adequate order parameter for this transition. A mean-field solution predicting first-order, critical and tricritical behaviours is analysed in detail. The second example refers to externally driven disordered systems that respond intermittently through avalanches. The interesting aspect is that for a critical amount of disorder, avalanches occur with an absence of characteristic scales, which define avalanche criticality as reported in different ferroic materials. This behaviour can be accounted for by lattice models with disorder, driven by athermal dynamics.

The chapter starts with a unified view of glassy states in ferroic materials. Disorder and frustration are the main ingredients responsible for the glassy behaviour, which is identified as a strong frequency dependence of the ac-susceptibility in addition to the occurrence of memory effects detected in zero-field-cooling (ZFC) versus field-cooling (FC) measurements of the temperature dependence of the main ferroic property. Dilute magnetic alloys are taken as prototypical examples of materials displaying glassy behaviour. The physical origin is justified by considering the random distribution of the low concentration of magnetic atoms and their RKKY oscillating exchange interaction. This behaviour is used to inspire lattice models which are (extensions of the Ising model) adequate to study glassy behaviour at a microscopic scale. The particular case of spin glasses is considered in detail and mean-field solutions based on the replica symmetry approach are discussed. Finally, similar models for relaxor ferroelectric and strain glasses are also introduced and briefly described.

Quantum phase transitions occur at zero temperature driven by quantum fluctuations instead of thermal fluctuations. They take place due to competing ground state phases that are accessible for different values of certain non-thermal parameters such as coupling constants, pressure or magnetic field. The chapter starts with a discussion of the main phenomenological features of this class of non-thermal transitions. In particular, it is argued that traces of these transitions can be detected at finite (but low) temperature. Then, examples of materials that show this behaviour are provided. Finally, the quantum Ising model is discussed and it is shown that a quantum model in d dimensions can be mapped to a classical model in d+1 dimensions.

The chapter discusses caloric materials, which are those that show large and reversible thermal response to an applied external field, either mechanical, electric or magnetic. The corresponding effects are denoted as mechanocaloric, electrocaloric and magnetocaloric effects, respectively. The response is usually quantified by the changes of entropy and temperature induced by isothermal and adiabatic application/removal of a field, respectively. These quantities are large in the vicinity of phase transitions and, in particular, close to a first-order transition where the latent heat provides a large caloric response. Well-known examples are ferroic materials in the vicinity of their transition towards the ferroic phase. The chapter starts with the study of caloric effects near a critical point and subsequently caloric behaviour near a first-order transition is analysed. Then, the possibility of multicaloric effects that can be induced by multiple fields in multiferroic materials is considered and a general thermodynamic formalism of multicaloric effects is developed.

Liquid crystals are complex materials that share properties of both solids and liquids. This is a consequence of complex anisotropic molecules that permit establishing phases with orientational and positional orders. Thus, a large variety of phases and phase transitions can occur in these systems. After a detailed description of general features of these materials, the tensorial nature of the orientational order parameter is discussed. Then, the Landau–de Gennes theory is developed for the isotropic–nematic transition. Later, positional degrees of freedom are included to account for the nematic–smectic transition. Next, the theory is generalized to include fluctuations, distortions and the effect of an external field. In the last part, topological defects are discussed with a particular emphasis on defects such as skyrmions and merons which can form in chiral liquid crystals such as cholesteric and blue phases. Finally, the analogy of these classes of defects with those occurring in non-collinear magnetic materials is considered.

The chapter introduces in a unified manner all ferroic materials including the three main ferroic systems, namely ferromagnetic, ferroelectric and ferroelastic, in addition to the case of materials that can display ferrotoroidic order. General physical aspects of magnetism, electricity and elasticity are used in order to introduce the order parameters that conveniently describe all these classes of ferroic phase transitions. It is shown that while the order parameter has a vectorial nature for ferromagnetic (axial vector), ferroelectric (polar vector) and magnetic ferrotoroidal (axial vector) systems, it is a rank-2 polar tensor in ferroelastic materials. The resulting physical differences arising from the different nature of the order parameter are then analysed in detail. Next, it is shown how to construct a convenient Ginszburg–Landau free energy functional in terms of these order parameters and their coupling for the different ferroic systems besides how to obtain the corresponding phase diagrams and microstructural features.