CONDENSED MATTER

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

S. N. Molotkov<sup>abc

a</sup> Institute of Solid State Physics, Russian Academy of Sciences
^b Academy of Criptography of Russia
^c M. V. Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics

Abstract: 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.

Received: 18.07.2013
Revised: 28.08.2013

DOI: 10.7868/S0370274X13190119

 English version: Journal of Experimental and Theoretical Physics Letters, 2013, 98:7, 410–415

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