Topological phases: isomorphism, homotopy and $K$-theory

TL;DR

This paper explores the mathematical classification of topological phases in condensed matter physics, focusing on the relationships between isomorphism, homotopy, and $K$-theory, especially in chiral systems where invariants are relative.

Contribution

It clarifies the subtle distinctions between different equivalence notions and reconciles them within the framework of $K$-theory for topological insulators.

Findings

Winding number in chiral systems is relative, not absolute.

$K$-theory provides a unified language for classifying topological phases.

Relations between isomorphism, homotopy, and $K$-theory are elucidated.

Abstract

Equivalence classes of gapped Hamiltonians compatible with given symmetry constraints, such as those underlying topological insulators, can be defined in many ways. For the non-chiral classes modelled by vector bundles over Brillouin tori, physically relevant equivalences include isomorphism, homotopy, and $K$-theory, which are inequivalent but closely related. We discuss an important subtlety which arises in the chiral Class AIII systems, where the winding number invariant is shown to be relative rather than absolute as is usually assumed. These issues are then analyzed and reconciled in the language of $K$-theory.