Classification of Interacting Topological Crystalline Superconductors in Three Dimensions and Beyond

Although classification for free-fermion topological superconductors (TSC) is established, systematically understanding the classification of 3D interacting TSCs remains difficult, especially those protected by crystalline symmetries like the 230 space groups. We build up a general framework for systematically classifying 3D interacting TSCs protected by crystalline symmetries together with discrete internal symmetries. We first establish a complete classification for fermionic symmetry protected topological phases (FSPT) with purely discrete internal symmetries, which determines the crystalline case via the crystalline equivalence principle. Using domain wall decoration, we obtain classification data and formulas for generic FSPTs, what are suitable for systematic computation. The four layers of decoration data $(n_1, n_2, n_3, \nu_4)$ characterize a 3D FSPT with symmetry $G_b\times_{\omega_2}Z_2^f$, corresponding to $p+ip$, Kitaev chain, complex fermion, and bosonic SPT layers. Inspired by previous works, a crucial aspect is the $p+ip$ layer, where classification involves two possibilities: anti-unitary and infinite-order symmetries (e.g., translation). We show the former maps to some mirror FSPT classification with the mirror plane decorated by a $p+ip$ superconductor, while the latter is determined by the free part of $H^1(G_b, Z_T)$, corresponding to weak TSCs. Another key point is the Kitaev chain decoration for the anti-unitary symmetries, which differs essentially from unitary ones. We explicitly obtain formulas for all three layers of decoration $(n_2, n_3, \nu_4)$, which are amenable to automatic computation. As an application, we classify the 230 space-group topological crystalline superconductors in interacting electronic systems.