Topological phases of non-interacting systems: A general approach based on states

TL;DR

This paper introduces a unified classification scheme for topological phases in non-interacting systems using homotopy classes of state configurations, encompassing known topological insulators within a broader mathematical framework.

Contribution

It develops a general approach based on states and homotopy classes for classifying topological phases, extending existing schemes to a wider class of systems.

Findings

Recovers the classification of non-interacting topological insulators of type A.

Provides a framework applicable to systems with trivial $C^*$-bundles.

Includes various examples illustrating the formalism.

Abstract

In this work we provide a classification scheme for topological phases of certain systems whose observable algebra is described by a trivial $C^*$-bundles. The classification is based on the study of the homotopy classes of \emph{configurations}, which are maps from a \emph{quantum parameter space} to the space of pure states of a reference \emph{fiber} $C^*$-algebra. Both the quantum parameter space and the fiber algebra are naturally associated with the observable algebra. A list of various examples described in the last section shows that the common classification scheme of non-interacting topological insulators of type A is recovered inside this new formalism.