Fermionic Symmetry Protected Topological Phases and Cobordisms

TL;DR

This paper tests cobordism theory as a classification tool for fermionic SPT phases with Z/2 symmetry, confirming known phases in low dimensions and predicting new interacting phases in higher dimensions.

Contribution

It validates cobordism classification for fermionic SPT phases up to dimension 3 and predicts the existence of new interacting phases in higher dimensions.

Findings

Cobordism correctly classifies all known fermionic SPT phases up to dimension 3.

All such phases in low dimensions can be realized by free fermions.

Prediction of inherently interacting fermionic SPT phases in higher dimensions.

Abstract

It has been proposed recently that interacting Symmetry Protected Topological (SPT) phases can be classified using cobordism theory. We test this proposal in the case of fermionic SPT phases with Z/2 symmetry, where Z/2 is either time-reversal or an internal symmetry. We find that cobordism classification correctly describes all known fermionic SPT phases in space dimension less than or equal to 3 and also predicts that all such phases can be realized by free fermions. In higher dimensions we predict the existence of inherently interacting fermionic SPT phases.