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.

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

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.

Quantum Hall states are the first examples of topological insulators that demonstrate completely contrasting electronic behavior between the bulk and the edges of the sample. The bulk of the system is insulating, while there exists conducting states at the edges. Moreover, the Hall conductivity is quantized in units of a universal constant, e2/h. It became clear later on that the quantization is actually related to a topological invariant known as the Chern number. The geometric interpretation of this invariant is provided by the Gauss–Bonnet theorem, which relates the integral of the Gaussian curvature over a closed surface to a constant that simply counts the number of ‘genus’ (or holes) of the object. In solid state physics, the closed surface is the Brillouin zone, and the Gaussian curvature is analogous to a quantity known as the Berry curvature, integral of which over the Brillouin zone yields the quantization of the Hall conductivity.

In Chapter 1, we begin with a historical overview of the quantum Hall effect. The experiment and the physical systems are described with an emphasis on the two-dimensional (2D) nature of the ‘dirty’ electronic system in the presence of a strong perpendicular magnetic field at low temperature. The Hall resistivity as a function of the field shows quantized plateaus in unit of h/e2 with an accuracy of one part in more than a billion. Very surprisingly, the longitudinal resistivity synergetically vanishes at the positions of the plateaus for the Hall resistivity. This indicates the emergence of a phase with an inherent ambiguousness of being a perfect conductor and a perfect insulator at the same time. However, such an ambiguity can only be reconciled for an electron gas confined in a plane in the presence of a magnetic field.

Quite intriguingly, the presence of the perpendicular magnetic field introduces ‘another’ quantization, which replaces the band structure (energy as a function of the wavevector) of the electronic system. This quantization was shown via solving the Schrödinger equation in the presence of a Landau gauge. The resultant energy levels of this problem are the infinitely degenerate Landau levels, which slightly broaden due to the presence of impurity and disorder but still remain distinct and cause quantization of the Hall conductivity as the magnetic field is ramped up gradually.

Introduction

Graphene is formed of C atoms. C is an element in the IVth column of the periodic table and has four valence electrons in the outermost shell. It can make two types of chemical bonds, namely sp3, which results in diamond known from ancient times, and a more stable sp2, which results in graphite that is known for the last 500 years. A quick look at the discoveries of different allotropes of C is available in Table 4.1. The sp2 hybridization causes planar configuration involving 3 of the 4 electrons, which are 120° apart and are bound by σ bonds that add stiffness (and flatness too) to the linkage between the C–C atoms, while the fourth electron bound to the C atoms via the π bond projects out of the plane, and is available for conduction. Thus, the electronic structure that we shall be discussing elaborately is due to these π electrons.

Graphene was the first discovery of atomically thin perfect two-dimensional (2D) material. Andre Geim and co-workers successfully exfoliated graphene from graphite [2, 3]. Some of the remarkable properties of graphene (which, unfortunately, we shall not worry too much about) include its strength, impermeability, very large thermal conductivity (at least one order larger than copper), as a molecule sensor, transparent (for its usage in displays), in the field of biology, such as neuron growth and DNA sequencing, and many more. Owing to the tremendous fundamental and technological applications of graphene, the discovery earned a Nobel Prize to A. Geim and K. Novoselov, both from the University of Manchester in the UK in 2010.

Introduction

The fractional quantum Hall effect (FQHE) was discovered by Tsui, Stormer and Gossard in 1982 at Bell Labs. They observed that at very high magnetic fields, a 2DEG shows fractional quantization of the Hall conductance. In particular, they got a quantized Hall plateau of magnitude ρxy = 3h/e2, which is accompanied by the vanishing of the longitudinal conductivity, ρxx, at low temperature (T < 5 K) in GaAs and AlGaAs samples. As opposed to the integer quantum Hall effect (IQHE), where an integer number of Landau levels (LLs) are occupied, here in FQHE the LLs are partially occupied. If onemakes themagnetic field large enough, the lowest Landau level (LLL) will be partially filled. Whatwe can expect is that the system will form some kind of a lattice, for example, a Wigner crystal or a charge density wave. Thus, it naively seems to be reasonable that the system would like to minimize its potential energy, since there is no kinetic energy left in the system corresponding to the LLL, and only a trivial zero point energy is present in the system. Thus, the ions tend to stay away from each other and form something similar to a crystal lattice. However, surprisingly that does not happen, and instead the system becomes an incompressible quantum liquid, which has gaps in the energy spectrum at filling 1/m (m: odd, or a rational fraction of the form n/m). So it is inevitable that the systemminimizes its energy by having gaps at fractional values of filling. The reason is that, owing to the presence of a large number of electrons (macroscopically degenerate in any of the LLs), a many-body interaction is induced, which in fact makes the excitations above this incompressible ground state to be fractional. So in essence, the Hall current carries a fractional charge.

Introduction

The date of discovery of the quantum Hall effect (QHE) is known pretty accurately. It occurred at 2:00 a.m. on 5 February 1980 at the high magnetic lab in Grenoble, France (see Fig. 1.1). There was an ongoing research on the transport properties of silicon field-effect transistors (FETs). The main motive was to improve the mobility of these FET devices. The devices that were provided by Dorda and Pepper allowed direct measurement of the resistivity tensor. The system is a highly degenerate two-dimensional (2D) electron gas contained in the inversion layer of a metal oxide semiconductor field effect transistor (MOSFET) operated at low temperatures and strong magnetic fields. The original notes appear in Fig. 1.1, where it is clearly stated that the Hall resistivity involves universal constants and hence signals towards the involvement of a very fundamental phenomenon.

In the classical version of the phenomenon discovered by E. Hall in 1879, just over a hundred years before the discovery of its quantum analogue, one may consider a sample with a planar geometry so as to restrict the carriers to move in a 2D plane. Next, turn on a bias voltage so that a current flows in one of the longitudinal directions and a strong magnetic field perpendicular to the plane of the gas (see Fig. 1.2). Because of the Lorentz force, the carriers drift towards a direction transverse to the direction of the current flowing in the sample. At equilibrium, a voltage develops in the transverse direction, which is known as the Hall voltage. The Hall resistivity, R, defined as the Hall voltage divided by the longitudinal current, is found to linearly depend on the magnetic field, B, and inversely on the carrier density, n, through R = B/nq (q is the charge). A related and possibly more familiar quantity is the Hall coefficient, denoted by RH = R/B, which via its sign yields information on the type of the majority carriers, that is, whether they are electrons or holes.

At very low temperature or at very high values of the magnetic field (or at both), the resistivity of the sample assumes quantized values of the form rxy = h/ne2. Initially, n was found to be an integer with extraordinary precession (one part in ∼ 108). This is shown in Fig. 1.3.