The cohomological nature of the Fu-Kane-Mele invariant

TL;DR

This paper extends the cohomological FKMM-invariant to classify topological phases in quantum systems beyond insulators, providing a unified framework and complete classifications for certain low-dimensional spaces.

Contribution

It generalizes the FKMM-invariant to broader settings and offers a full classification of Quaternionic vector bundles over specific low-dimensional involutive manifolds.

Findings

Cohomological description of the Fu-Kane-Mele index.

Generalization of the FKMM-invariant to non-insulator quantum systems.

Complete classification of Quaternionic vector bundles over low-dimensional spheres and tori.

Abstract

In this paper we generalize the definition of the FKMM-invariant introduced in [DG2] for the case of "Quaternionic" vector bundles over involutive base spaces endowed with free involution or with a non-finite fixed-point set. In [DG2] it has already be shown how the FKMM-invariant provides a cohomological description of the Fu-Kane-Mele index used to classify topological insulators in class AII. It follows that the FKMM-invariant described in this paper provides a cohomological generalization of the Fu-Kane-Mele index which is applicable to the classification of protected phases for other type of topological quantum systems (TQS) which are not necesarily related to models for topological insulators (e.g. the two-dimensional models of adiabatically perturbed systems discussed in [GR]). As a byproduct we provide the complete classification of "Quaternionic" vector bundles over a big class of low dimensional involutive spheres and tori.