Topological Band Theory, An Overview

Abstract: Topological insulators, superconductors and semi-metals are states of ma er with unique features such as quantized macroscopic observables and robust, gapless edge states. ese states can not be explained by standard quantum mechanics, but require also the framework of topology to be properly characterized. Topology is a branch of mathematics having to do with properties that are conserved under continuous deformations of spaces. is review presents some of the ways in which topology and condensed ma er physics come together, with a focus on non-interacting models which can be described with a band theory approach. Furthermore, the focus is on insulating systems but the discussions may sometimes be applied to superconductors and semi-metals. e eld of topological phases of ma er is not all together new, yet it lacks elementary introductions to newcomers. is review is meant for those with basic condensed ma er physics background and aims at providing a self-consistent overview of the central concepts in the eld of topological ma er. e structure of the review is as follows: In Chapter 1, a brief historical background is given. Also, a basic introduction to topology is presented, with focus on how it is used in condensed ma er physics. Following this, Chapter 2 introduces three important discrete symmetries which are key in characterizing topological phases of ma er. In particular, the e ect that these symmetries have on a general Bloch Hamiltonian is shown. In Chapter 3, the e ect of discrete symmetries on certain models is investigated. e well-known Su-Schrie er-Heeger model is discussed because it is the simplest models known to exhibit a topological phase and a topological invariant. Chapter 4 broadens the discussion of this topological invariant which is a winding number. Chapter 5 introduces the geometric phase (Berry phase) which is used to describe another topological invariant, the Chern number, the subject of Chapter 6. ere the alternative interpretations of the Chern number are discussed. A erwards, in Chapter 7, the quantum Hall e ect is presented. Following this, a general classi cation scheme for topological phases of fermionic, non-interacting systems will be presented in Chapter 8. It will be shown how it can be determined whether a system could possibly host a topological phase or not based on the symmetries of the Hamiltonian. Chapter 9 focuses on the concepts pertaining to the physics of the gapless edge states which appear between the interface of a (non-interacting) topological insulator and a topologically trivially insulator. Among the concepts discussed here is the bulk boundary correspondence and topological protection. Lastly, Chapter 10 contains a brief recap of what has been established in the review and some conclusionary remarks.