Zigzag edge modes in Z2 topological insulator: reentrance and completely flat spectrum

The spectrum and wave function of helical edge modes in Z_2 topological insulator are derived on a square lattice using Bernevig-Hughes-Zhang (BHZ) model. The BHZ model is characterized by a "mass" term M (k) that is parameterized as M (k) = Delta - B k^2. A topological insulator realizes when the parameters Delta and B fall on the regime, either 0 < Delta /B < 4 or 4 < Delta /B < 8. At Delta /B = 4, which separates the cases of positive and negative (quantized) spin Hall conductivities, the edge modes show a corresponding change that depends on the edge geometry. In the (1,0)-edge, the spectrum of edge mode remains the same against change of Delta /B, although the main location of the mode moves from the zone center for Delta /B < 4, to the zone boundary for Delta /B > 4 of the 1D Brillouin zone. In the (1,1)-edge geometry, the group velocity at the zone center changes sign at Delta /B = 4 where the spectrum becomes independent of the momentum, i.e. flat, over the whole 1D Brillouin zone. Furthermore, for Delta/B < 1.354..., the edge mode starting from the zone center vanishes in an intermediate region of the 1D Brillouin zone, but reenters near the zone boundary, where the energy of the edge mode is marginally below the lowest bulk excitations. On the other hand, the behavior of reentrant mode in real space is indistinguishable from an ordinary edge mode.