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.

In this chapter, we shall discuss three paradigmatic models that show symmetry-protected topological features and are resilient to local perturbations as long as the relevant symmetries are not disturbed. They are Su–Schrieffer–Heeger (SSH) model and a Kitaev chain with superconducting correlations in one-dimensional (1D) and a ladder system, known as the Creutz ladder in a quasi-1D setup.

Su—Schrieffer—Heeger (SSH) Model

Introduction

To make our concepts clear on the topological phase, and whether a model involves a topological phase transition, we apply it to the simplest model available in the literature. The SSH model denotes a paradigmatic 1D model that hosts a topological phase. It also possesses a physical realization in polyacetylene, which is a long chain organic polymer (polymerization of acetylene) with a formula [C2H2]n (shown in Fig. 3.1). The C–C bond lengths are measured by NMR spectroscopy technique and are found to be 1.36 Å and 1.44 Å for the double and the single bonds respectively. The chain consists of a number of methyne (= CH−) groups covalently bonded to yield a 1D structure, with each C-atom having a p electron. This renders connectivity to the polymer chain.

Possibly intrigued by this bond-length asymmetry, one can write down a tight-binding Hamiltonian of such a system with two different hopping parameters for spinless fermions hopping along the single and the double bonds. These staggered hopping amplitudes are represented by t1 and t2. Let us consider that the chain consists of N unit cells with two sites (that is, two C atoms) per unit cell and denote these two sites as A and B. The hopping between A and B sites in a cell be denoted by t1, while those from B to A across the cell can be denoted by t2. Because of the presence of a single π electron at each of the C atoms, the interparticle interaction effects are completely neglected. We shall show that the staggered hopping or the dimerization has got serious consequences for the topological properties of even such a simple model.

Chapter 2 is a gentle introduction to the many-body physics of polyacetylene. Band theory for electrons hopping along a one-dimensional lattice is explained. The continuum limit is taken. An example of a quantum critical point with its emergent symmetries is given. The Su–Schrieffer–Heeger (SSH) model for polyacetylene is defined and solved at the mean-field level.