Zigzag edge modes in aZ2topological insulator: Reentrance and completely flat spectrum

Abstract

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.

Edge modes in zigzag and armchair ribbons of monolayer MoS2

“…The low-energy Hamiltonian of this system is not exactly a massive Dirac model, as it includes a momentum dependent mass (∆ + βq 2 ) similar to the modified Dirac model of a topological insulator thin film [34] and to the HgTe-based quantum spin Hall system (BHZ-model) [35]. Some of the topological aspects of a modified-Dirac system have been explored after discretizing the low-energy model on a square lattice using a finite difference (FD) method [36].…”

Topological Insulator and the Dirac Equation

“…Thus for a finite B, the band gap may not open at the Γ point in the lattice model because of the competition between the linear term and the quadratic term of k i . This fact may lead to a topological transition from a large B to a small B. Imura et al [64] analyzed the 2D case in details and found that there exists a topological transition at a finite value of B in two-dimension. A similar transition will also exist in higher dimension.…”

Helical edge modes preserved across transitions from topological insulators with indirect gaps to insulators or semimetals

“…Helical edge states have been further studied using the resultant continuum model [12,13] and the lattice regularized model [14][15][16]. In these previous papers, the particle-hole symmetry has been assumed.…”