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

Abstract

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.

19 pages, 33 figures

References in corpus (12)

• The electronic properties of graphene
• Quantum Spin Hall Effect in Graphene
• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Topological Insulators with Inversion Symmetry
• The Quantum Spin Hall Effect: Theory and Experiment
• Massive Dirac fermions and spin physics in an ultrathin film of topological insulator
• Finite size effects of helical edge states in HgTe/CdTe quantum wells
• Effective continuous model for surface states and thin films of three dimensional topological insulators
• Band structure of semimagnetic Hg1-yMnyTe quantum wells
• Quantum spin Hall phases

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• Analytic Theory of Edge Modes in Topological Insulators
• Generic helical edge states due to Rashba spin-orbit coupling in a topological insulator
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• Edge states of a three-dimensional topological insulator
• Continuum Theory of Edge States of Topological Insulators: Variational Principle and Boundary Conditions
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