The spectrum and wave function of helical edge modes in Z2 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) = ∆ − Bk 2 . A topological insulator realizes when the parameters ∆ and B fall on the regime, either 0 < ∆/B < 4 or 4 < ∆/B < 8. At ∆/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 ∆/B, although the main location of the mode moves from the zone center for ∆/B < 4, to the zone boundary for ∆/B > 4 of the 1D Brillouin zone. In the (1, 1)-edge geometry, the group velocity at the zone center changes sign at ∆/B = 4 where the spectrum becomes independent of the momentum, i.e. flat, over the whole 1D Brillouin zone. Furthermore, for ∆/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.
Spectrum and wave function of gapless edge modes are derived analytically for
a tight-binding model of topological insulators on square lattice. Particular
attention is paid to dependence on edge geometries such as the straight (1,0)
and zigzag (1,1) edges in the thermodynamic limit. The key technique is to
identify operators that combine to annihilate the edge state in the effective
one-dimensional (1D) model with momentum along the edge. In the (1,0) edge, the
edge mode is present either around the center of 1D Brillouin zone or its
boundary, depending on location of the bulk excitation gap. In the (1,1) edge,
the edge mode is always present both at the center and near the boundary.
Depending on system parameters, however, the mode is absent in the middle of
the Brillouin zone. In this case the binding energy of the edge mode near the
boundary is extremely small; about $10^{-3}$ of the overall energy scale.
Origin of this minute energy scale is discussed.Comment: 9pages, 7figur
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