On a symmetry topological classification of edge states in crystalline spin-Hall insulators with the time reversal invariance

Symmetry analysis reveals all types of singularities of the edge states in two-dimensional systems with a boundary (2D $\to$ 1D systems), which are invariant under time reversal. Symmetry reasons also provide the matching condition for material functions parameterizing the Hamiltonian at various points of the Brillouin zone. The unified parameterization of the Hamiltonian makes it possible to construct the mapping of trajectories closed in the quasimomentum $k$ in the Brillouin zone into the $SU(2)$ topological group. There are only two equivalence classes of Hamiltonians, which are given by the elements of the first fundamental group $\pi_1(SO(3))=\pi_1(SU(2)/{\mathcal Z}_2)={\mathcal Z}_2$. The first type of surface states corresponds to a normal insulator and the second type corresponds to a topological spin-Hall insulator. Comparison with the ${\mathcal Z}_2$ classification based on the Pfaffian method is performed.