Interacting SPT phases are not Morita invariant
Abstract
Class D topological superconductors have been described as invertible topological phases protected by charge $Q$ and particlehole symmetry $C$. A competing description is that class D has no internal symmetries except for the fermion parity group $\mathbb{Z}_2^F = \{1, (1)^F\}$. In the weakly interacting setting, it can be argued that `particlehole symmetry cancels charge' in a suitable sense. Namely, the classification results are independent of which of the two internal symmetry groups are taken because of a Morita equivalence. However, we argue that for strongly interacting particles, the group of symmetryprotected topological phases in the two cases are nonisomorphic in dimension $2+1$. This shows that in contrast to the free case, interacting phases are not Morita invariant. To accomplish this, we use the approach to interacting phases using invertible field theories and bordism. We give explicit expressions of invertible field theories which have the two different groups $\mathbb{Z}_2^F$ and $U(1)_Q \rtimes \mathbb{Z}_2^C$ as internal symmetries and give spacetime manifolds on which their partition functions disagree. Techniques from algebraic topology are used to compute the relevant bordism groups, most importantly the James spectral sequence. The result is that there are both a new $\mathbb{Z}_2$ and a new $\mathbb{Z}$invariant for $U(1)_Q \rtimes \mathbb{Z}_2^F$ that are not present for $\mathbb{Z}_2^F$.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.07408
 Bibcode:
 2021arXiv211007408S
 Keywords:

 High Energy Physics  Theory;
 Condensed Matter  Strongly Correlated Electrons;
 Mathematical Physics;
 Mathematics  Algebraic Topology;
 81T45 (Primary) 55N22;
 82D03;
 57R15;
 57R56 (Secondary)
 EPrint:
 22 pages, 1 figure