Competition between on-site and off-site pairing phases and surface spectra in superconducting wallpaper fermion systems

We theoretically investigate the competition between on-site and off-site pairings in superconducting topological crystalline insulators hosting wallpaper fermions, a class of surface states protected by nonsymmorphic symmetries. In-plane nearest-neighbor pair hopping stabilizes the ${\mathrm{A}}_{1\mathrm{u}}$ phase, while off-site pairing channels yield additional dominant phases with ${\mathrm{B}}_{2\mathrm{u}}$ and ${\mathrm{E}}_{\mathrm{u}}$ representations. We further show that the ${\mathrm{B}}_{2\mathrm{u}}$ pairing state supports the hybridization between wallpaper-fermion spectra themselves, whereas a representative time-reversal-invariant nematic ${\mathrm{E}}_{\mathrm{u}}$ state exhibits spectral hybridization between wallpaper fermion and double Majorana Kramers pairs, yielding twisted surface energy spectra. These results demonstrate that the twisted dispersions arising from hybridization phenomena persist in the presence of off-site pairing interactions.