3. - Topology in One-Dimensional (1D) and Quasi-1D Models

Summary

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.