Index of Dirac operators and classification of topological insulators

TL;DR

This paper uses Dirac operators and index theorems on Clifford bundles to derive the periodic table of topological insulators and superconductors, linking topological invariants to algebraic structures.

Contribution

It provides a mathematical framework connecting Clifford algebra, K-theory, and index theorems to classify topological phases of matter.

Findings

Topological invariants are systematically derived from Clifford algebra.

The periodic table of topological insulators is explained via index theorems.

Clifford chessboard underpins the classification scheme.

Abstract

Real and complex Clifford bundles and Dirac operators defined on them are considered. By using the index theorems of Dirac operators, table of topological invariants is constructed from the Clifford chessboard. Through the relations between K-theory groups, Grothendieck groups and symmetric spaces, the periodic table of topological insulators and superconductors is obtained. This gives the result that the periodic table of real and complex topological phases is originated from the Clifford chessboard and index theorems.