Interacting SPT phases are not Morita invariant

TL;DR

This paper demonstrates that interacting topological phases in certain symmetry classes are sensitive to the specific symmetry group chosen, challenging the Morita invariance assumption in the classification framework.

Contribution

It introduces two mathematical interpretations of class D symmetry, showing they lead to different classifications of interacting topological phases in 2+1 dimensions.

Findings

Morita-equivalent symmetry groups can yield different interacting classifications.

K-theory cannot distinguish between different symmetry group interpretations.

Interacting classifications are sensitive to the exact symmetry group used.

Abstract

The tenfold way provides a strong organizing principle for invertible topological phases of matter. Mathematically, it is intimately connected with $K$-theory via the fact that there exist exactly ten Morita classes of simple real superalgebras. This connection is physically unsurprising, since weakly interacting topological phases are classified by $K$-theory. We argue that when strong interactions are present, care has to be taken when formulating the exact ten symmetry groups present in the tenfold way table. We study this phenomenon in the example of class D by providing two possible mathematical interpretations of a class D symmetry. These two interpretations of class D result in Morita-equivalent but different symmetry groups. As $K$-theory cannot distinguish Morita-equivalent protecting symmetry groups, the two approaches lead to the same classification of topological phases on the weakly interacting side. However, we show that these two different symmetry groups yield different interacting classifications in spacetime dimension 2+1. We use the approach to interacting topological phases using bordism groups, reducing the relevant classification problem to a spectral sequence computation.