Superconducting gap structures in wallpaper fermion systems

TL;DR

The paper addresses how wallpaper fermions on the surface of nonsymmorphic topological crystalline insulators gap out when subjected to superconductivity. It adopts a two-dimensional effective model to classify six symmetry-allowed, momentum-independent pair potentials and analyzes their gap structures using zero-dimensional topological invariants and group-theoretical (Mackey-Bradley) methods. The main results show that Delta2 hosts a point node, while Delta5 and Delta6 host line nodes, with distinct protection mechanisms: some nodes are protected by Z2 invariants at momentum points or lines, and others are enforced by crystalline symmetry on glide-related lines. This work clarifies how nonsymmorphic symmetry shapes superconducting gaps and highlights the potential for gapless wallpaper fermion states and related quasiparticles on the surface.

Abstract

We theoretically investigate the superconducting gap structures in wallpaper fermions, which are surface states of topological nonsymmorphic crystalline insulators, based on a two-dimensional effective model. A symmetry analysis identifies six types of momentum-independent pair potentials. One hosts a point node, two host line nodes, and the remaining three are fully gapped. By classifying the Bogoliubov--de Gennes Hamiltonian in the zero-dimensional symmetry class, we show that the point and line nodes are protected by $\mathbb{Z}_2$ topological invariants. In addition, for the twofold-rotation-odd pair potential, nodes appear on the glide-invariant line and are protected by crystalline symmetries, as clarified by the Mackey--Bradley theorem.

Figures (11)

• Figure 1: (Color online) Crystal structure protecting wallpaper fermions. This crystalline symmetry corresponds to the wallpaper group $\mathrm{p4g}$. Two glides $\{m_{01}|1/2\ 1/2\}$ and $\{m_{10}|1/2\ 1/2\}$ are indicated by the dashed lines.
• Figure 2:
• Figure 4: (Color online) Nodal structures in superconducting wallpaper fermions. We identify the momentum points where the energy is smaller than $10^{-4}$ as gapless points. We set $\mu=0.1$ and assume the weak-coupling pairing ($\Delta_0=0.005$). The other parameters are the same values used in Fig. \ref{['fig:wp_energy']}.
• Figure 11: (Color online) Fermi surface and the number of occupied bands at each momentum points $N_\mathrm{occ}(\bm{k})$.
• Figure 12: