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The spectrum and wave function of helical edge modes in Z_2 topologicalinsulator are derived on a square lattice using Bernevig-Hughes-Zhang (BHZ)model. The BHZ model is characterized by a "mass" term M (k) that isparameterized as M (k) = Delta - B k^2. A topological insulator realizes whenthe 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 andnegative (quantized) spin Hall conductivities, the edge modes show acorresponding change that depends on the edge geometry. In the (1,0)-edge, thespectrum of edge mode remains the same against change of Delta /B, although themain location of the mode moves from the zone center for Delta /B < 4, to thezone boundary for Delta /B > 4 of the 1D Brillouin zone. In the (1,1)-edgegeometry, the group velocity at the zone center changes sign at Delta /B = 4where the spectrum becomes independent of the momentum, i.e. flat, over thewhole 1D Brillouin zone. Furthermore, for Delta/B < 1.354..., the edge modestarting from the zone center vanishes in an intermediate region of the 1DBrillouin zone, but reenters near the zone boundary, where the energy of theedge mode is marginally below the lowest bulk excitations. On the other hand,the behavior of reentrant mode in real space is indistinguishable from anordinary edge mode.

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