Zigzag edge modes in a 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)=Delta-Bk-2. A topological insulator realizes when the parameters Delta and B fall on the regime, either 0 4 of the one-dimensional (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.