Topological nonsymmorphic crystalline insulators

In this work, we identify a class of topological phases protected by nonsymmorphic crystalline symmetry dubbed “topological nonsymmorphic crystalline insulators.” We construct a concrete tight-binding model for a lattice with nonsymmorphic symmetry and confirm its topological nature by directly calculating topological surface states. Analogous to “Kramers' pairs” originating from time-reversal symmetry, we introduce “doublet pairs” originating from nonsymmorphic symmetry to define the corresponding $Z_2$ topological invariant for this phase. Based on projective representation theory, we extend our discussion to other nonsymmorphic symmetry groups that can host this topological phase which will provide guidance for the systematic search for new topological materials.

Figure 1

(a) Lattice sites in the $xz$ plane for our tight-binding model. The lattice vectors ${\vec{a}}_1$ and ${\vec{a}}_3$ are shown in the figure and ${\vec{a}}_2$ is perpendicular to this plane $(y$ direction). (b) The bulk Brillouin zone (BZ) and the surface BZ for the $xz$ plane.

Figure 2

Energy dispersion of a slab configuration for our tight-binding model with the parameter (a) $M_1=3.1$ and (b) $M_1=4.5$. The corresponding evolution of Wannier function centers in a 3D bulk system is shown as a function of $k_x$ for (c) $M_1=3.1$ and (d) $M_1=4.5$. The surface states in (b) and the winding number of Wannier function centers in (d) indicate that the system is in the TNCI phase for $M_1=4.5$.

Figure 3

(a) Schematic plot of a semi-infinite crystal with one surface. (b) Energy dispersion of nontrivial surface states. Here ${\vec{K}}_1$ and ${\vec{K}}_2$ are two HSM.

Figure 4

Energy dispersion of a slab configuration at different $M_1$ values: (a) $M_1=3.1<3.4$; the system is topologically trivial. (b) $3.4<M_1=4.5<5$; the system is topologically nontrivial with a single gapless surface state at $\overline{Z}$. (c) $5<M_1=5.8<6.6$; the system is topologically nontrivial with single gapless surface state at $\overline{U}$ instead of $\overline{Z}$. (d) $M_1=7>6.6$; system is topologically trivial.

Figure 5

Wannier center flows for our tight-binding model characterizing three topological phase transitions (TPTs): First TPT at $M_1=3.4$: (a) $M_1=3.39$ (winding number = 0); (b) $M_1=3.41$ (winding number = 1). Second TPT at $M_1=5$ (this TPT shows no change in winding number because the band gap closes twice at the same time): (c) $M_1=4.99$ (winding number = 1); (d) $M_1=5.01$ (winding number = 1). Third TPT at $M_1=6.6$: (e) $M_1=6.59$ (winding number = 1); (f) $M_1=6.61$ (winding number = 0).

Figure 6

(a) A structure with $pgg$ symmetry in the $xz$ plane. Two lattice vectors are ${\vec{a}}_1=(a,0,0)$ and ${\vec{a}}_2=(0,0,c)$, and the vector $\vec{\tau }=(\frac{a}{2},0,\frac{c}{2})$. $A$ and $B$ atoms are denoted by red and blue colors. Taking the point between two adjacent $B$ atoms as the origin, two generators of the $pgg$ group are $\left\{{\widehat{m}}_x\right|\vec{\tau }\}$ and $\left\{{\widehat{m}}_z\right|\vec{\tau }\}$. (b) A structure with $p4g$ symmetry in the $xz$ plane. Two lattice vectors are ${\vec{a}}_1=(a,0,a)$ and ${\vec{a}}_2=(-a,0,a)$, and the vector $\vec{\tau }=(a,0,0)$. Two generators of the $p4g$ group are $\left\{C_4\right|\vec{e}\}$ and $\left\{{\widehat{m}}_x\right|\vec{\tau }\}$.