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

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.

Figures (57)

• Figure 1: Cell decomposition for the point group $SG=C_4$. The 2D system (left) is decomposed into symmetry-related blocks (right). Any two-dimensional block $\sigma$ is mapped to any other $2$D block through a finite number of $\pi/2$ rotations. The $1$D blocks $\tau$ are also connected by $\pi/2$ rotations.
• Figure 2: Cell decompositions for cyclic groups $C_n$ with $n=2,3,4,6$, arranged from left to right. Each cell decomposition contains a single symmetrically independent block in each dimension ($0$D, $1$D, and $2$D), denoted as $\mu$, $\tau$, and $\sigma$, respectively.
• Figure 3: Cell decompositions for dihedral groups $D_n$ with $n=2,3,4,6$, arranged from left to right. Each cell decomposition contains one symmetrically independent block for dimensions $0$D and $2$D, denoted as $\mu$ and $\sigma$, respectively. However, there are two classes of independent $1$D blocks, labeled by $\tau_1$ and $\tau_2$.
• Figure 4: 1D $G_f$-FSPT constructed by two Majorana chains. The arrows indicate pairings between Majorana operators (depicted as blue dots); for example, an arrow pointing from $\gamma^L_{j+1}$ to $\gamma^R_j$ represents a coupling term $i\gamma^R_{j} \gamma^L_{j+1}$. The two chains are labeled by index $A$ and $B$, respectively.
• Figure 5: Single Majorana chain decoration on $1$D blocks labeled by $\tau_1$ or $\tau_2$, leaving $4$ dangling Majorana modes (depicted as blue dots) $\gamma_1,\gamma_2,\gamma_3,\gamma_4$ at $0$D block $\mu$.