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.