Real-space construction and classification for time-reversal symmetric crystalline superconductors in 2D interacting fermionic systems

TL;DR

This paper classifies 2D interacting fermionic topological superconductors with crystalline and time-reversal symmetries, identifying intrinsic phases with unique real-space constructions and higher-order boundary modes.

Contribution

It introduces a systematic real-space classification of crystalline topological superconductors, revealing intrinsically interacting phases and their explicit constructions.

Findings

Intrinsic $Z_4$ classified phases for specific wallpaper groups.

Real-space decorated 1D fermionic SPT block constructions.

Discovery of higher-order FSPT phases with corner zero modes.

Abstract

Crystalline symmetry and time-reversal symmetry are commonly present in real superconducting materials. However, the topological classification of systems respecting these symmetries, particularly for interacting fermions, remains incomplete. In this work, we systematically classify time-reversal symmetry-protected crystalline topological superconductors in two-dimensional interacting fermionic systems using an explicit real-space construction. Among the resulting phases, we identify intrinsically interacting fermionic topological superconductors, i.e., phases that cannot be realized in either free-fermion or interacting bosonic systems. For spinless fermions with protecting symmetry group $C_4 \times Z_2^T$ or $D_4 \times Z_2^T$ (plus fermion parity), the intrinsic sector has a $Z_4$ classification. The corresponding root phases generating this $Z_4$ classification admit a transparent real-space construction in terms of decorated 1D blocks. These blocks are 1D fermionic symmetry-protected topological (FSPT) phases, realizable as double Majorana chains. We further find the corresponding $Z_4$ spinless intrinsic phases for wallpaper groups $p4$, $p4m$, and $p4g$. We also find an additional $Z_2$ intrinsically interacting phase for spinless fermions with wallpaper group $pm$, which is absent with the corresponding point-group symmetry alone. Moreover, these intrinsic phases naturally give rise to higher-order FSPT phases that support corner zero modes. Finally, we verify the crystalline equivalence principle for generic 2D interacting FSPT systems with both crystalline and internal symmetries.