Controlling edge states in the Kane-Mele model via edge chirality

We investigate the dependence of band dispersion of the quantum spin Hall effect (QSHE) edge states in the Kane-Mele model on crystallographic orientation of the edges. Band structures of the one-dimensional honeycomb lattice ribbons show the presence of the QSHE edge states at all orientations of the edges given sufficiently strong spin-orbit interactions. We find that the Fermi velocities of the QSHE edge-state bands increase monotonically when the edge orientation changes from zigzag (chirality angle $θ= 0^\circ$) to armchair ($θ= 30^\circ$). We propose a simple analytical model to explain the numerical results.