Topological insulators from the perspective of non-commutative geometry and index theory

TL;DR

This paper reviews recent mathematical advances in understanding topological insulators using non-commutative geometry and index theory, highlighting their topological invariants and surface states.

Contribution

It provides a non-technical overview of how index theory and non-commutative geometry elucidate the topological properties of insulators.

Findings

Topological invariants characterize non-trivial insulators.

Surface states are robust against localization.

Mathematical structures explain topological phases.

Abstract

Topological insulators are solid state systems of independent electrons for which the Fermi level lies in a mobility gap, but the Fermi projection is nevertheless topologically non-trivial, namely it cannot be deformed into that of a normal insulator. This non-trivial topology is encoded in adequately defined invariants and implies the existence of surface states that are not susceptible to Anderson localization. This non-technical review reports on recent progress in the understanding of the underlying mathematical structures, with a particular focus on index theory.