Bott--Kitaev periodic table and index theory

TL;DR

This paper clarifies the index theory behind Kitaev's periodic table for topological insulators and superconductors, focusing on spatial dimensions one to three and organizing invariants via KR-theory.

Contribution

It provides a rigorous mathematical framework connecting the periodic table to index theory and KR-theory, emphasizing the analytical and topological invariants in different dimensions.

Findings

First Chern number characterizes 2D invariants.

$bZ_2$ invariants are computed via odd topological index.

Framework unifies topological invariants using KR-theory.

Abstract

We consider topological insulators and superconductors with discrete symmetries and clarify the relevant index theory behind the periodic table proposed by Kitaev. An effective Hamiltonian determines the analytical index, which can be computed by a topological index. We focus on the spatial dimensions one, two and three, and only consider the bulk theory. In two dimensions, the $\mathbb{Z}$-valued invariants are given by the first Chern number. Meanwhile, $\mathbb{Z}_2$-valued invariants can be computed by the odd topological index and its variations. The Bott-Kitaev periodic table is well-known in the physics literature, we organize the topological invariants in the framework of KR-theory.