Topology of nonsymmorphic crystalline insulators and superconductors

Topological classification in our previous paper [K. Shiozaki and M. Sato, Phys. Rev. B 90, 165114 (2014)] is extended to nonsymmorphic crystalline insulators and superconductors. Using the twisted equivariant $K$ theory, we complete the classification of topological crystalline insulators and superconductors in the presence of additional order-two nonsymmorphic space-group symmetries. The order-two nonsymmorphic space groups include half-lattice translation with ${\mathit{Z}}_2$ flip, glide, twofold screw, and their magnetic space groups. We find that the topological periodic table shows modulo-2 periodicity in the number of flipped coordinates under the order-two nonsymmorphic space group. It is pointed out that the nonsymmorphic space groups allow ${\mathbb{Z}}_2$ topological phases even in the absence of time-reversal and/or particle-hole symmetries. Furthermore, the coexistence of the nonsymmorphic space group with time-reversal and/or particle-hole symmetries provides novel ${\mathbb{Z}}_4$ topological phases, which have not been realized in ordinary topological insulators and superconductors. We present model Hamiltonians of these new topological phases and analytic expressions of the ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_4$ topological invariants. The half-lattice translation with ${\mathit{Z}}_2$ spin flip and glide symmetry are compatible with the existence of boundaries, leading to topological surface gapless modes protected by the order-two nonsymmorphic symmetries. We also discuss unique features of these gapless surface modes.

Figure 1

A $(d+1)$-dimensional bulk TCI/TCSC and its $d$-dimensional boundary.

Figure 2

Schematic illustration of order-two nonsymmorphic ${\mathit{Z}}_2$ symmetry in two dimensions. The system is invariant under half-lattice translation in the $x$ direction followed by the exchange of two different (blue and red) states. A one-dimensional edge parallel to the $x$ direction is compatible with this symmetry. Gapless edge states protected by this symmetry appear on this edge when the relevant bulk topological number is nonzero.

Figure 3

An order-two nonsymmorphic ${\mathit{Z}}_2$ symmetry in three dimensions. The system is invariant under half-lattice translation in the $x$ direction followed by the exchange of blue and red planes. A two-dimensional surface parallel to the $x$ direction is compatible with this symmetry. Gapless surface states protected by this symmetry appear on this surface when the relevant bulk topological number is nonzero.

Figure 4

Gapless edge states in two-dimensional time-reversal-breaking TCSCs with nonsymmorphic ${\mathit{Z}}_2$ symmetry $G\left(k_x\right)$ ($d_{\parallel }=0, d=2, U$ in class D). $G{\left(k_x\right)}^2=e^{-i{k}_x}$. In (a)–(c) [(d)–(f)], the gap function is even (odd) under $G\left(k_x\right)$. “$+$” and “$-$” in (a)–(c) [(d–(f)] indicate the eigenvalue $g_{\pm }\left(k_x\right)$ of $G_{\mathrm{BdG}}^{(+)}\left(k_x\right) \left[G_{\mathrm{BdG}}^{(-)}\left(k_x\right)\right]$ for edge states.

Figure 5

Side view of the square lattice. In the presence of the staggered modulation in the right figure, the size of the unit cell is doubled in the $x$ direction. The right lattice configuration has a nonsymmorphic ${\mathit{Z}}_2$ symmetry: turn it upside down and move it by half of the unit cell in the $x$ direction, then the system goes back to the original configuration.

Figure 6

(a) [(b)] A typical edge spectrum for $\theta =2$ ($\theta =1$). (c) The integral region for the ${\mathbb{Z}}_4$ invariant (47). The left- and right-hand-side regions represent the positive and negative glide eigensectors, respectively, which are glued together on their boundary lines $k_x=-\pi ,\pi$. In figure (c), the red circles at $(k_x,k_y)=(0,0)$ and $(0,\pi )$ are the centers of the particle-hole transformation, and the blue circles at $(k_x,k_y)=(\pi ,0)$ and $(\pi ,\pi )$ are the centers of the time-reversal transformation. The integral region for the ${\mathbb{Z}}_4$ invariant (47) is the shaded portion in (c).

Figure 7

Gapless edge states in two-dimensional time-reversal-invariant TCSCs with nonsymmorphic ${\mathit{Z}}_2$ symmetry $G^{(-)}\left(k_x\right)$ ($d_{\parallel }=0, d=2, \{U,\mathrm{\Gamma }\}=0$ in class DIII). Here, $G^{(-)}{\left(k_x\right)}^2=-e^{-i{k}_x}$. The gap function is odd under the nonsymmorphic ${\mathit{Z}}_2$ symmetry. Red ($+$) and blue ($-$) lines indicate edge states in the $g_{\pm }\left(k_x\right)$ sector. (a) Edge states in the $\theta =1$ phase. (b), (c) Edge states in the $\theta =2$ phase. (d) Edge states in the $\theta =3$ phase. Figure (e) [(f)] shows adiabatic transformation of edge states in the $\theta =2$ ($\theta =3$) phase. The modulo-4 ambiguity of $\theta$ is evident in (e) and (f).

Figure 8

Gapless edge states in two-dimensional time-reversal-invariant TCSCs with nonsymmorphic ${\mathit{Z}}_2$ symmetry $G\left(k_x\right)$ ($d_{\parallel }=0, d=2, [U,\mathrm{\Gamma }]=0$ in class DIII). $G{\left(k_x\right)}^2=-e^{-i{k}_x}$. The gap function is even under $G\left(k_x\right)$. (a) Edge states in the $({\nu }_{+},{\nu }_{-})=(1,0)$ phase. (b) Edge states in the $({\nu }_{+},{\nu }_{-})=(0,1)$ phase. (c) Edge states in the $({\nu }_{+},{\nu }_{-})=(1,1)$ phase. Red ($+$) and blue ($-$) lines indicate edge states with the eigenvalues $g_{\pm }\left(k_x\right)=\pm i{e}^{-i{k}_x/2}$ of $G_{\mathrm{BdG}}^{(+)}\left(k_x\right)$. Figure (c) also shows an adiabatic deformation of the edge states.

Figure 9

Schematic illustration of glide symmetry in three dimensions. The red and blue indicate two different configurations, say spin configuration, that are exchanged by mirror reflection with respect to the $zx$ plane. The system is invariant under glide reflection with respect to the $zx$ plane, i.e., combination of mirror reflection and half-lattice translation in the $x$ direction. A two-dimensional surface parallel to the $x$ direction is compatible with the glide symmetry. Gapless surface states protected by glide symmetry appear on the glide-symmetric surface when the relevant bulk topological number is nonzero.

Figure 10

Schematic pictures of ${\mathbb{Z}}_2$ nontrivial surface states with glide symmetry ($d_{\parallel }=1, d=3, U$ in class A). Figures (a) and (b) show surface states with the same nontrivial ${\mathbb{Z}}_2$ number. The green volumes represent the bulk spectrum. In (b), the red curves are identified at the zone boundary.

Figure 11

Schematic pictures of ${\mathbb{Z}}_4$ nontrivial surface states with glide symmetry ($d_{\parallel }=1, d=3, U$ in class AII). The green volumes represent the bulk spectrum. (a) ${\mathbb{Z}}_4=1$ ($\theta =1$) phase with single a surface Dirac fermion. (b), (c) ${\mathbb{Z}}_4=2$ ($\theta =2$) phase. Two Dirac fermions (b) can be adiabatically transformed into a detached surface state (c). In (c), the red curves are identified at the zone boundary.

Figure 12

Adiabatic deformation of gapless surface states in the ${\mathbb{Z}}_4=2$ ($\theta =2$) phase. The figures show the energy spectra at the glide-invariant line $k_y=0$. Red ($+$) and blue ($-$) lines show surface states in the $g_{\pm }\left(k_x\right)=\pm i{e}^{-i{k}_x/2}$ sectors.

Figure 13

Hamiltonian raising map from $T^d$ to $T^d\times S^1$.