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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann its first Chern number. The total Chern number of the Kane–Mele model is always zero, since for a time reversal invariant ground state c1↑ = −c1↓ and hence c1 = c1↑ + c1↓ = 0. It is, however, possible to define a new topological invariant, i.e. the spin Chern number 2cspin := c1↑ − c1↓. (3.10) In general, the Rashba coupling is non-zero, then the spin-up and spin-down components are coupled together and the definition of the spin Chern number is no longer valid. However, there always exists a Z2 invariant for the quantum spin Hall effect, for example, the Kane–Mele invariant discussed in Sec. 4.1. If a topological insulator can be specified by a Z2 invariant, we call it a Z2 topological insulator. The quantum spin Hall effect was also predicted by Bernevig, Hughes and Zhang [17] for the HgTe/CdTe quantum well with stronger spin-orbit interaction, which was observed by Ko ̈nig et al. [52] soon after the prediction. The effective Hamilto- nian for this so-called quantum well is given by where HBHZ = H(k) 0 0 H∗(−k) H(k) = Akxσx + Akyσy + (M − Bk2)σz + (C − Dk2)σ0 (3.11) here A, B, C, D are expansion parameters and M is a mass or gap parameter. Notice that this effective Hamiltonian, up to the σ0-term — which actually drops out in the quantum Hall response —, is of the form discussed in Sec. 2.3. This system again decouples, so that the simpler version of the Z2 invariant given by cspin is applicable. 3.1.4. 3d topological insulators Later on, 3d topological insulators were also observed in nature. In 3d the role of edge states is played by codimension-1 surface states. The observed surface states look like double cones in the spectrum or energy in the momentum dispersion relation in physical terms. They take the form of local conical singularities x2 +y2 = z2, and are called Dirac cones by physicists, see e.g., [46]. A point in the Brillouin torus over which a Dirac cone lies is called a Dirac point. In 3d topological insulators, a Dirac cone is sometimes also referred to as a Majorana zero mode. Note that a Majorana zero mode should not be confused with a Majorana fermion. If the system is invariant under time reversal symmetry then the so-called Kramers degeneracy forces level sticking at any fixed point of the time reversal symmetry, as discussed in the next section. If these singularities are not extended 1630003-16 Rev. Math. Phys. 2016.28. Downloaded from www.worldscientific.com by PURDUE UNIVERSITY on 08/11/17. For personal use only.PDF Image | Notes on topological insulators
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