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.

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.

Chapter 5 introduces a family of exactly soluble spin-1/2 lattice Hamiltonians: The spin-1/2 cluster “c” chains. Each member of this family is gapped and nondegenerate when periodic boundary conditions hold. The nondegeneracy of the ground state is lifted for all members of this family except for one member, owing to the presence of zero modes bound to the boundaries when open boundary conditions hold. The notion of symmetry fractionalization is thereby introduced. This family of exactly soluble Hamiltonians is mapped of a family of exactly soluble Majorana lattice Hamiltonians, one of which is an example of a Kitaev chain, though the Jordan–Wigner transformation. The stability of the degeneracy of the zero modes to integrability-breaking but symmetry-preserving interactions is derived through the explicit construction of the stacking rules.

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.

Chapter 4 aims at establishing that the fractional charge calculated in Chapter 3 is sharp. To this end, the calculation of the mean value and second cumulant of the electronic charge localized in one of two wells of a double-well potential in quantum mechanics is contrasted to that of the mean value and second cumulant of the fractional charge localized around a soliton in a dimerization profile of polyacetylene support a pair of soliton and anti-soliton defects far apart from each other.

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.

Chapter 8 extends the 10-fold way of gapped phases from one to any dimension of space. This is done by presenting the homotopy groups of the classifying spaces of normalized Dirac masses. There follows two applications. First, there is the interplay between Anderson localization and the topology of classifying spaces for disordered quantum wires. Second, it is possible to derive the breakdown of the 10-fold way due to short-range interactions in any dimension. The chapter closes with the relationship between invertible topological phases and invertible topological field theories.

Chapter 1 describes how the book is organized.