Classification of Interacting Topological Crystalline Superconductors in Three Dimensions and Beyond

TL;DR

This work develops a comprehensive framework to classify three-dimensional interacting topological superconductors protected by discrete internal and crystalline symmetries by combining domain-wall decoration (DWD) with the fermionic crystalline equivalence principle (CEP). It resolves key obstructions, notably the higher obstruction $O_5$ and a new Majorana-dimer obstruction $O_5^{\gamma}$ arising from anti-unitary symmetries, and provides explicit formulas for the four decoration layers $(n_1,n_2,n_3,\nu_4)$, including the $p+ip$ layer's dependence on $H^1(G_b,\mathbb{Z}_T)$ and translations. By mapping crystalline problems to internal symmetry classifications via CEP and employing an Atiyah-Hirzebruch spectral sequence (AHSS) interpretation of the domain-wall decorations, the authors obtain the complete interacting classification for all 230 space groups and illustrate the method with concrete examples. The results yield new insights into weak indices from translations, Majorana-dimer phases, and the role of anti-unitary structure in obstructing or permitting certain decorations, with implications for correlated materials and potential quantum simulators. The framework provides a scalable path toward systematic databases of interacting topological crystalline superconductors and connects constructive fixed-point models to abstract cohomological classifications.

Abstract

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 $(n_1, n_2, n_3, ν_4)$ characterize a 3D FSPT with symmetry $G_b\times_{ω_2}Z_2^f$, 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 $(n_2, n_3, ν_4)$, which are amenable to automatic computation. As an application, we classify the 230 space-group topological crystalline superconductors in interacting electronic systems.

Figures (12)

• Figure 1: (a). The chiral or antichiral Majorana fermions of $p+ip$ ($p-ip$) superconductor decoration on the dual planes of the three links of the triangle $\langle 123\rangle$ meet at the link dual to the triangle. If $n_1(12)>0$ ($n_1(12)<0$), then there is $|n_1(12)|$ gapless Majorana fermions propagating into (out of) the triangle. (b) The chiral fermions meet in a tetrahedron.
• Figure 2: 3D $G^{\mathrm{m}}_\mathrm{f}$ FSPT from $p+ip$ superconductor state decorated on the mirror plane. On the mirror plane, the total symmetry is $G^{\mathrm{m}}_\mathrm{f}$, whether the $p+ip$ state is compatible with $G^{\mathrm{m}}_\mathrm{f}$ is fully determined by the fact whether $\omega_2^{\mathrm{m}}=\omega_2+s_1\cup s_1$ is coboundary or not.
• Figure 3: The resolved dual lattice. The black tetrahedra in (a) is the positive oriented with the red in (a) sketch the resolved dual lattice. (b) only present the resolved dual lattice corresponding to the (black) tetrahedra in (a), while (c) is the simplified version of (b) for convenience. (d)-(f) is for the negative oriented.
• Figure 4: All possibilities of Kitaev chain decoration within a single 3-simplex. The solid red line with arrow indicates there is a Majorana pairing, while the dotted red line with arrow indicates no Majorana pairing (just drawn as reference). (a) Vacuum state, also called reference state, in which all $n_2 = 0$. (b) $n_2(012) = n_2(013) = 1$, while all other $n_2 = 0$. (c-g) Similar as (b), only one non-trivial Majorana pairing within the 3-simplex. (h) All $n_2 = 1$, two non-trivial Majorana pairing within the 3-simplex. We take the convention that $\gamma_{012A}$ is paired with $\gamma_{023A}$ while $\gamma_{013B}$ is paired with $\gamma_{123B}$.
• Figure 5: Transition loop with 2 and 4 sites