A Short Course on Topological Insulators: Band-structure topology and edge states in one and two dimensions

TL;DR

This paper provides a concise, accessible introduction to topological insulators in one and two dimensions, focusing on edge states, topological invariants, and the bulk-boundary correspondence using simple models.

Contribution

It offers a beginner-friendly, step-by-step explanation of topological band insulators, emphasizing mathematical simplicity and practical models like SSH and BHZ.

Findings

Illustrates edge states and topological invariants in simple models

Explains bulk-boundary correspondence in topological insulators

Provides educational material with problems for self-study

Abstract

This course-based primer provides newcomers to the field with a concise introduction to some of the core topics in the emerging field of topological band insulators in one and two dimensions. The aim is to provide a basic understanding of edge states, bulk topological invariants, and of the bulk--boundary correspondence with as simple mathematical tools as possible. We use noninteracting lattice models of topological insulators, building gradually on these to arrive from the simplest one-dimensional case (the Su-Schrieffer-Heeger model for polyacetylene) to two-dimensional time-reversal invariant topological insulators (the Bernevig-Hughes-Zhang model for HgTe). In each case the model is introduced first and then its properties are discussed and subsequently generalized. The only prerequisite for the reader is a working knowledge in quantum mechanics, the relevant solid state physics background is provided as part of this self-contained text, which is complemented by end-of-chapter problems.