A noncommutative framework for topological insulators

TL;DR

This paper introduces a noncommutative index theory approach to topological insulators, using Kasparov theory to unify the classification of topological phases via operator algebras and symmetries.

Contribution

It formulates the topological invariants of insulators within a noncommutative index framework, connecting physical symmetries to Kasparov modules and spectral triples.

Findings

Real and complex index pairings reproduce the periodic table of topological insulators.

Kasparov theory provides a unified mathematical framework for symmetry classification.

Noncommutative geometry captures the topology of possibly noncommutative Brillouin zones.

Abstract

We study topological insulators, regarded as physical systems giving rise to topological invariants determined by symmetries both linear and anti-linear. Our perspective is that of noncommutative index theory of operator algebras. In particular we formulate the index problems using Kasparov theory, both complex and real. We show that the periodic table of topological insulators and superconductors can be realised as a real or complex index pairing of a Kasparov module capturing internal symmetries of the Hamiltonian with a spectral triple encoding the geometry of the sample's (possibly noncommutative) Brillouin zone.