Symmetries, Dimensions, and Topological Insulators: the mechanism behind the face of the Bott clock

TL;DR

This paper explores the mathematical foundations of Bott periodicity and their application to the electronic properties of topological insulators and related materials, linking abstract topology to condensed matter physics.

Contribution

It connects advanced mathematical concepts like Bott periodicity to the physical understanding of topological insulators, offering new insights into their electronic band structures.

Findings

Mathematical framework for topological insulators

Application of Bott periodicity to electron band theory

Insights into the topology of spin Hall systems

Abstract

We provide an account of some of the mathematics of Bott periodicity and the Atiyah, Bott, Shapiro construction. We apply these ideas to understanding the twisted bundles of electron bands that underly the properties of topological insulators, spin Hall systems, and other topologically interesting materials.