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.

It is somewhat implicit that the readers are familiar with the first course on solid state physics, which mainly deals with electronic systems and teaches us how to distinguish between different forms of matter, such as metals, semiconductors and insulators. An elementary treatise on band structure is introduced in this regard, and in most cases, interacting phenomena, such as magnetism and superconductivity, are taught. The readers are encouraged to look at the classic texts on solid state physics, such as the ones by Kittel, Ashcroft and Mermin.

As a second course, or an advanced course on the subject, more in-depth study of condensed matter physics and its applications to the physical properties of various materials have found a place in the undergraduate curricula for a century or even more. The perspective on teaching the subject has remained unchanged during this period of time. However, the recent developments over the last few decades require a new perspective on teaching and learning about the subject. Quantum Hall effect is one such discovery that has influenced the way condensed matter physics is taught to undergraduate students. The role of topology in condensed matter systems and the fashion in which it is interwoven with the physical observables need to be understood for deeper appreciation of the subject. Thus, to have a quintessential presentation for the undergraduate students, in this book, we have addressed selected topics on the quantum Hall effect, and its close cousin, namely topology, that should comprehensively contribute to the learning of the topics and concepts that have emerged in the not-so-distant past. In this book, we focus on the transport properties of two-dimensional (2D) electronic systems and solely on the role of a constant magnetic field perpendicular to the plane of a electron gas. This brings us to the topic of quantum Hall effect, which is one of the main verticals of the book. The origin of the Landau levels and the passage of the Hall current through edge modes are also discussed. The latter establishes a quantum Hall sample to be the first example of a topological insulator. Hence, our subsequent focus is on the subject topology and its application to quantum Hall systems and in general to condensed matter physics. Introducing the subject from a formal standpoint, we discuss the band structure and topological invariants in 1D.

In this chapter, we shall discuss the interplay of symmetry and topology that are essential in understanding the topological protection rendered by the inherent symmetries and how the topological invariants are related to physical quantities.

Introduction

Poincaré conjecture was the first conjecture made on topology which asserts that a three-dimensional (3D) manifold is equivalent to a sphere in 3D subject to the fulfilment of a certain algebraic condition of the form f (x, y, z) = 0, where x, y and z are complex numbers. G. Perelman has (arguably) solved the conjecture in 2006 [4]. However, on practical aspects, just the reverse of what Poincaré had predicted happened. Topology and its relevance to condensed matter physics have emerged in a big way in recent times. The 2016 Nobel Prize awarded to D. J. Thouless, J. M. Kosterlitz, F. D. M. Haldane and C. L. Kane and E. Mele getting the Breakthrough Prize for contribution to fundamental physics in 2019 bear testimony to that.

Topology and geometry are related, but they have a profound difference. Geometry can differentiate between a square from a circle, or between a triangle and a rhombus; however, topology cannot distinguish between them. All it can say is that individually all these shapes are connected by continuous lines and hence are identical. However, topology indeed refers to the study of geometric shapes where the focus is on how properties of objects change under continuous deformation, such as stretching and bending; however, tearing or puncturing is not allowed. The objective is to determine whether such a continuous deformation can lead to a change from one geometric shape to another. The connection to a problem of deformation of geometrical shapes in condensed matter physics may be established if the Hamiltonian for a particular system can be continuously transformed via tuning of one (or more) of the parameter(s) that the Hamiltonian depends on. Should there be no change in the number of energy modes below the Fermi energy during the process of transformation, then the two systems (that is, before and after the transformation) belong to the same topology class. In the process, something remains invariant. If that something does not remain invariant, then there occurs a topological phase transition.

Chapter 7 opens with the description of superconductivity in terms of Bogoliubov–de-Gennes Hamiltonians. The 10-fold way in terms of the Altland–Zirnbauer symmetry classes is applied to random matrix theory and two disordered quantum wires. The chapter closes with the 10-fold way for the gapped phases of quantum wires.

Chapter 6 ties invertible topological phases to extensions of the original Lieb–Schultz–Mattis theorem. A review is made of the original Lieb–Schultz–Mattis theorem and how it has been refined under the assumption that a continuous symmetry holds. Two extensions of the Lieb–Schultz–Mattis theorem are given that apply to the Majorana chains from Chapter 5 when protected by discrete symmetries. To this end, it is necessary to introduce the notion of projective representations of symmetries and their classifications in terms of the second cohomology group. A precise definition is given of fermionic invertible topological phases and how they can be classified by the second cohomology group in one-dimensional space. Stacking rules of fermionic invertible topological phases in one-dimensional space are explained and shown to deliver the degeneracies of the boundary states that are protected by the symmetries.

Traditionally the different states of matter are described by symmetries that are broken. Typical situations include the freezing of a liquid, which breaks the translational symmetry that the fluid possessed, and the onset of magnetism, where the rotational symmetry is broken by the ordering of the individual magnetic moment vectors. In the early eighties of the previous century a completely new organizational principle of quantum matter was introduced following the discovery of the quantum Hall effect. The robustness of the quantum Hall state was a forerunner of the variety of topologically protected states that forms a large fraction of the condensed matter physics and material science literature at present.

Given the rapid strides that this field has made in the last two decades, it is almost imperative that it should become a part of the senior undergraduate curriculum. This necessitates the existence of a textbook that can address these somewhat esoteric topics at a level which is understandable to those who have not yet decided to specialize in this particular field but very well could, if given a proper exposition. This is a rather difficult task for the author of a textbook of a contemporary topic, and this is where the present book is immensely successful.

I am not a specialist in this subject by any means and found the book to be a comprehensive introduction to the area. I am sure the senior undergraduates and the beginning graduate students will benefit immensely from the book.

Jayanta K. Bhattacharjee

School of Physical Sciences,

Indian Association for the Cultivation of Science

Jadavpur, Kolkata

Chapter 3 is devoted to fractionalization in polyacetylene. Topological defects (solitons) in the dimerization of polyacetylene are introduced and shown to bind electronic zero modes. The fractional charge of these zero modes is calculated by different means: (1) The Schrieffer counting formula(2) Scattering theory(3) Supersymmetry(4) Gradient expansion of the current(5) Bosonization.

The effects of temperature on the fractional charge and the robustness of the zero modes to interactions in polyacetylene are studied.