Hourglass fermion surface states in stacked topological insulators with nonsymmorphic symmetry

Recently, a nonsymmorphic topological insulator was predicted, where the characteristic feature is the emergence of an “hourglass fermion” surface state protected by the nonsymmorphic symmetry. This prediction was supported soon after by the angle-resolved photoemission spectroscopy. We propose a simple model possessing the hourglass fermion surface state. The model is constructed by stacking the quantum spin Hall insulators with the interlayer coupling introduced so as to preserve the nonsymmorphic symmetry and the time-reversal symmetry. The Dirac theory is also derived, whose analytical results reproduce the hourglass fermion surface state remarkably well. Furthermore, we discuss how the hourglass state is destroyed by introducing perturbations based on the symmetry analysis. Our results show that the hourglass fermion surface state is universal in the helical edge system with the nonsymmorphic symmetry.

Figure 1

Band structure of the glided QAH insulator model. Bird's-eye views of the energy spectrum for the surface state in the $(k_x,k_z)$ plane for (a1) $t_{+}{t}_{-}<0$, (a2) $t_{+}{t}_{-}>0$, (a3) $t_{+}=0$, and (a4) $t_{-}=0$. (b1)–(b4) The band structure along the $\mathrm{\Gamma }Z$ line corresponding to (a1)–(a4), i.e., along the line $k_x=0$ for $0\le c{k}_z\le \pi$. The red curves near the Fermi energy represent the surface state. (c1)–(c4) The energy spectrum based on the Dirac theory corresponding to (b1)–(b4). The eigenvalues of the glide operator are shown. (d) The Brillouin zone for the surface, where $X=(\pi /a,0), \mathrm{\Gamma }=(0,0), Z=(0,\pi /c)$, and $U=(\pi /a,\pi /c)$ in the $(k_x,k_z)$ plane. Here, $a$ and $c$ are the lattice constants of the honeycomb lattice and the $k_z$ direction, respectively.

Figure 2

Band structure of the glided QSH insulator. (a) Bird's-eye views of the energy spectrum for the surface state in the $(k_x,k_z)$ plane. There are two vertically placed Dirac cones at the $\mathrm{\Gamma }$ point, representing the hourglass fermion surface state. The nonsymmorphic symmetry protected gap closing point is marked by a green circle. The Kramers (pseudo-Kramers) doublets are marked by cyan (magenta) circles. (b) The band structure along the $X\mathrm{\Gamma }ZU$ line. The red curves near the Fermi energy represent the surface state. (c) The energy spectrum based on the Dirac theory corresponding to (b), reproducing well the results for the surface state. The band crossing on the $\mathrm{\Gamma }Z$ line is protected by the nonsymmorphic symmetry, while the band crossing on the $X\mathrm{\Gamma }$ line originates in helical edges. (d) The Brillouin zone for the surface. (e) The band structure along the $k_x$ direction for various $k_z$.

Figure 3

Band structure of the glided QSH insulator model for $t_{+}{t}_{-}=0$. (a1), (a2) Bird's-eye views of the energy spectrum of the surface state for (a1) $t_{+}=0$, and (a2) $t_{-}=0$. (b1), (b2) The band structure along the $X\mathrm{\Gamma }ZU$ line corresponding to (a1) and (a2). The red curves near the Fermi energy represent the surface state.

Figure 4

Hourglass fermion surface state without the TRS. The TRS is broken by applying the magnetic field. Bird's-eye views of the energy spectrum of the surface state for (a1) $B_x\ne 0$, (a2) $B_y\ne 0$, and (a3) $B_z\ne 0$. (b1)–(b3) The energy spectrum along the $X\mathrm{\Gamma }ZU$ line corresponding to (a1)–(a3). The surface band structure becomes different between both sides of the edges. That of the one side is colored in magenta, while that of the other side is colored in cyan.

Figure 5

Energy spectrum of the hourglass fermion with the perturbation $V$ based on the Dirac theory. The band crossings protected by the glide symmetry $G_x$ occur for $V\propto 1, {\sigma }_x, {\sigma }_y{\eta }_z, {\sigma }_z{\eta }_z$ and are marked by solid green circles, while those which are not protected by $G_x$ are marked by dotted green circles. The energy degeneracies protected by the time-reversal symmetry $T$ occur at the $\mathrm{\Gamma }$ point for $V\propto 1, {\eta }_x, {\sigma }_x{\eta }_y, {\sigma }_y{\eta }_y, {\sigma }_z{\eta }_y, {\eta }_z$ and are denoted by solid cyan circles, while those which are not protected by $T$ are marked by the dotted cyan circles. The energy degeneracies protected by the local chiral symmetry $C_0$ occurs for $V\propto {\eta }_y, {\sigma }_x{\eta }_y, {\sigma }_y{\eta }_y, {\sigma }_z{\eta }_y, {\eta }_z, {\sigma }_x{\eta }_z, {\sigma }_y{\eta }_z, {\sigma }_z{\eta }_z$ and are marked by solid cyan squares. The energy degeneracies at the $Z$ point protected by the time-reversal nonsymmorphic symmetry $G_{x}T$ occurs for $V\propto 1, {\sigma }_y, {\sigma }_z, {\sigma }_x{\eta }_z$ are marked by solid magenta circles, while those which are not protected by $G_{x}T$ are marked by dotted magenta circles. The energy degeneracies protected by the local chiral symmetry $C_{\pi }$ occur for $V\propto {\sigma }_y, {\sigma }_z, {\eta }_x, {\sigma }_x{\eta }_x, {\sigma }_y{\eta }_y, {\sigma }_z{\eta }_y, {\eta }_z, {\sigma }_x{\eta }_z$ and are marked by solid magenta squares. The degeneracies occur due to the fact $V\propto H_{\text{gQSH}}$ at the $\mathrm{\Gamma }$ point for ${\eta }_x$ and at the $Z$ point for ${\sigma }_x{\eta }_y$, and are marked by solid purple squares. Symmetric band structures along $E=0$ due to the chiral symmetry for $V\propto {\sigma }_y, {\sigma }_x{\eta }_x, {\eta }_y, {\sigma }_z{\eta }_z$ are marked by solid purple diamonds.