Topological classification of crystalline insulators with space group symmetry

We show that in crystalline insulators, space group symmetry alone gives rise to a topological classification based on the discretization of electric polarization. Using $C_3$ rotational symmetry as an example, we first prove that the polarization is discretized into three distinct classes, i.e., it can only take three inequivalent values. We then prove that these classes are topologically distinct. Therefore, a $Z_3$ topological classification exists, with polarization as a topological class index. A concrete tight-binding model is derived to demonstrate the $Z_3$ topological phase transition. Using first-principles calculations, we identify graphene on a BN substrate as a possible candidate to realize these $Z_3$ topological states. To complete our analysis, we extend the classification of band structures to all 17 two-dimensional space groups. This work will contribute to a complete theory of symmetry-conserved topological phases and also elucidate topological properties of graphenelike systems.

Figure 1

Topological phase diagram of the tight-binding model described by the Hamiltonian (7). The solid (blue) curve is the phase boundary given by ${\Delta }_{\alpha }{\Delta }_{\beta }=t^{\prime 2}$. In each region, the polarization is given by $P_{\mathrm{I}}=-(\vec{a}+\vec{b})/3$, $P_{\mathrm{II}}=0$, and $P_{\mathrm{III}}=(\vec{a}+\vec{b})/3$. The dashed line indicates a representative path in the parameter space that undergoes the topological phase transition twice.

Figure 2

(a), (b) Schematic illustration of polarization in systems with $C_3$ symmetry. (a) $h$-BN lattice with three equivalent unit cells. (b) Possible values of polarization. Each topological class is marked by a specific symbol. Parts (c)–(f) display calculated values of polarization for different structures and the corresponding topological classes. (c) Structure 1: $h$-BN monolayer; (d) structure 2: $AA$ stacking of graphene monolayer on $h$-BN; (e) $AB$ stacking of graphene monolayer on $h$-BN with a carbon sublattice lying above the boron sublattice; and (f) structure 4: $AB$ stacking with a carbon sublattice lying above the nitrogen sublattice. The unit cells and lattice vectors $\vec{a}$ and $\vec{b}$ are marked in black, whereas the polarization vector is marked in red. Colors used to represent various atoms are listed.

Figure 3

Allowed values of polarization for all 17 two-dimensional space groups. The corresponding Wigner-Seitz cells for various lattices are displayed in blue. Allowed points and lines of points are marked in magenta. Allowed values of polarization are vectors joining the origin of the Wigner-Seitz cells to the points marked.