Double-twisted surface spectrum from hybridized Majorana Kramers pairs and wallpaper fermions

Abstract

We theoretically investigate the superconducting surface states of wallpaper fermions, which are surface quasiparticles of topological nonsymmorphic crystalline insulators protected by a wallpaper group $p4g$ symmetry, based on a tight-binding model for the space group $P4/mbm$ (No. 127). A symmetry-based analysis shows that four types of on-site pair potentials are allowed. Using the symmetries of the wallpaper group $p4g$ and the one-dimensional topological invariants, we clarify that for the $\mathrm{A_{1u}}$ representation, wallpaper fermions and two Majorana Kramers pairs coexist, and hybridization between them give rise to a double-twisted surface state and produces four peaks in the surface density of states. We further find that the mirror Chern number vanishes, indicating that our system realizes mirror-helicity-free surface states. This distinguishes superconducting wallpaper fermions from the other superconducting topological (crystalline) insulators, such as $\mathrm{Cu}_x\mathrm{Bi}_2\mathrm{Se}_3$ and $\mathrm{Sn}_{1-x}\mathrm{In}_x\mathrm{Te}$.

Figures (6)

• Figure 1: (Color online) (a) Tetragonal lattice model for wallpaper fermions wieder2018wallpaper. (b) Brillouin zone for space group $P4/mbm$ and (001) surface Brillouin zone Aroyo2011-crAroyo2006-bi1Aroyo2006-bi2Aroyo2014brillouin.
• Figure 2: (Color online) Surface energy spectrum for a semi-infinite system with the (001) surface on the C--D layers. The inset shows an enlarged view near the $\bar{\mathrm{M}}$ point. A fourfold degeneracy at the $\bar{\mathrm{M}}$ point and a twofold degeneracy along the $\bar{\mathrm{X}}\bar{\mathrm{M}}$ line are confirmed. The parameters are fixed as $t_1=1,\ v_{\mathrm{r}1}=0.25,\ v_{\mathrm{s}1}=0.2,\ t_2=0.5,\ v_{\mathrm{s}2}=-0.2,\ v'_{\mathrm{s}2}=0.15,\ u_1=0.3,\ u_2=-0.45,\ w_1=0.2,\ w_2=0.05,\ w_3=0.05,\ w_4=-0.02,\ w_5=0.11,\ w_6=-0.08,\ v_1=0.2,\ v_2=0.1,\ p_1=0,\ p_2=0.3$.
• Figure 3: (Color online) Surface spectral function for a (001) surface on the C--D layer along the $\bar{\Gamma}\bar{\mathrm{M}}\bar{\Gamma}$ line in (a) the normal state and (b) the superconducting state with $\Delta_3$ pairing ($\Delta_0=0.02,\mu=-0.75$). (c) Surface and (d) bulk density of states for the $\Delta_3$ pairing.
• Figure 4: (Color online) Schematic surface dispersions for (a) a topological insulator, (b) its superconducting state, (c) a topological crystalline insulator, and (d) its superconducting state. Solid green (dotted magenta) lines indicate the dispersions belonging to $+i$ ($-i$) mirror eigenspace.
• Figure 5: (Color online) Energy spectra along the $\bar{\Gamma}\bar{\mathrm{M}}\bar{\Gamma}$ line, resolved into mirror eigensectors. Normal state spectra in the (a) $+i$ and (b) $-i$ sectors. Superconducting state spectra for the $\Delta_3$ pairing in the (c) $+i$ and (d) $-i$ sectors.