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A topological insulator has a remarkable property of being metallic on the surface albeit insulating in the bulk. Recently much focus is on a specific type of topological insulators, which is said to be Z2-nontrivial, and realized as a consequence of interplay between spin-orbit coupling and a band structure with effective relativistic dispersion (electrons satisfy a Dirac equation). Such systems are invariant under time reversal and shows Kramers degeneracy.
Graphene is a two-dimensional (2D) implementation of such Dirac electron system, which also realizes a prototype [1] of Z2 topological insulator, in the presence of intrinsic and extrinsic (Rashba) spin-orbit interactions. Recently, we were able characterize specific localization properties of this system, in the presence of weak disorder and under doping [2]. We have shown, in particular, that localization symmetry class is determined by the parity of the total number Ns of "activated" effective spins in the system. Our diagnosis provides a contemporary version of the weak localization theory. The correspondence between the physics of bulk and of the edge highlights the physics of a topological insulator [3]. To demonstrate, we make a close comparison between the continuum Dirac theory and the square lattice tight-binding model [4] for a Z2-topological insulator in 2D. It is naturally shown that not only explicit, but also hidden Dirac cones in the high-energy spectrum, contribute to and ensure the integral quantization of spin Hall conductance. The nature of gapless edge modes in the lattice model under different (straight vs. zigzag) boundary conditions is extensively studied. We demonstrate that the edge modes of a Z2-topological insulator are susceptible of various finite size effects, in comparison with chiral edge modes of quantum Hall insulator. By comparing the behavior of gapless edge modes in real and momentum spaces, we show that localizability of edge mode is intricately related to level crossings due to projection of 2D bulk spectrum onto the 1D edge. [1] C.L. Kane and E.J. Mele, Phys. Rev. Lett. 95, 146802 (2005); ibid., 226801. [2] K.-I. Imura, Y. Kuramoto and K. Nomura, arXiv:0904.1676, Europhys. Lett. 89, 17009 (2010); arXiv:0907.5051, Phys. Rev. B 80, 085119 (2009). [3] K.-I. Imura, A. Yamakage, A. Hotta and Y. Kuramoto, in preparation; K.-I. Imura, Shijun Mao, A. Yamakage and Y. Kuramoto, in preparation. [4] B.A. Bernevig, T.L. Hughes and S.-C. Zhang, Science 314, 1757 (2006). |
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