Interacting SPT phases are not Morita invariant

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.

arXiv comment25 pages, 1 figure, v2: major revision of first part based on discussions with an anonymous referee, v3: minor revision for publication in Letters in Mathematical Physics

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Cited by in corpus (4)

• Rigorous Index Theory for One-Dimensional Interacting Topological Insulators
• The Smith Fiber Sequence and Invertible Field Theories
• Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders
• Smith homomorphisms and Spin${}^h$ structures