Superconducting Gap Structures in Wallpaper Fermion Systems

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.

💡 Research Summary

This paper investigates the superconducting gap structures that emerge on the surface states—so‑called wallpaper fermions—of topological nonsymmorphic crystalline insulators (TCIs). Wallpaper fermions appear on the (001) surface of TCIs possessing the p4g wallpaper group, which contains two orthogonal glide symmetries and a fourfold rotation. The authors construct a two‑dimensional k·p effective Hamiltonian (Eq. 1) that respects all space‑group operations and time‑reversal symmetry. The Hamiltonian is a 4 × 4 matrix acting in spin (s) and sublattice (σ) space, with five real parameters v₁…v₅ that reproduce the characteristic fourfold degeneracy at the (\bar M) point and twofold degeneracy along the (\bar X)–(\bar M) line.

Superconductivity is introduced via a Bogoliubov–de Gennes (BdG) Hamiltonian (Eq. 2). The pairing potential (\hat\Delta_i) is assumed momentum‑independent (weak‑coupling limit) and constrained by Fermi‑Dirac statistics ((\hat\Delta_i = s_y \hat\Delta_i^t s_y)). By decomposing the pairing into irreducible representations of the point group C₄ᵥ, six symmetry‑distinct pairings are identified (Table 2): Δ₁–Δ₆, belonging respectively to A₁, A₂, B₁, B₂, and the two components of the E representation. Each pairing carries a definite parity under the glide and rotation operations.

Numerical diagonalization of the BdG Hamiltonian (with Δ₀ = 0.005, μ = 0) reveals three distinct gap scenarios: (i) fully gapped spectra for Δ₁, Δ₃, and Δ₄; (ii) a point node at the (\bar M) point for Δ₂; (iii) line nodes along high‑symmetry directions for Δ₅ (along kₓ) and Δ₆ (along k_y). The authors then ask why certain nodes are protected while others are not.

To answer this, they treat each momentum point as a zero‑dimensional (0D) system and classify the BdG Hamiltonian according to Altland–Zirnbauer symmetry classes. Time‑reversal (˜Θ), particle‑hole (˜Ξ), and chiral (˜Γ) symmetries are combined with a spatial operation d that maps k → –k, yielding effective 0D operators ˜Θ₀, ˜Ξ₀, and ˜Γ₀ (Eqs. 9–11). The square of d’s representation, D(d)², together with the parity χ(d) of the pairing under d, determines the 0D class (Table 3). When χ(d)=–1 and D(d)²=–1, the system belongs to class BDI, which carries a ℤ₂ invariant ν_k