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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann that the analytical index of H ̃e equals the negative of the spectral flow of the edge states, inda(H ̃e) = −ν. The topological index of H ̃e was also computed explicitly [36] by investigating the divergence of a chiral current of H ̃e, see loc. cit. for details. The topological index turns out to be the difference of the first Chern numbers specifying the two bulk ground states separated by the domain wall, −indt(H ̃e) = c1(M+) − c1(M−). By the Atiyah–Patodi–Singer index theorem, the analytical index equals the topological index for H ̃e and hence by the generalized index theorem ν = c1(M+) − c1(M−) (2.20) which can be interpreted as the bulk-edge correspondence between the two bulk states and the edge states. A physical picture is as follows: by stacking two systems with topologically distinct ground states, an edge state is created at the interface. The transport property of the edge state is then determined by the topology of the junction. This would also work for edge states in the quantum spin Hall effect, see e.g., [41]. 3. Z2 Topological Insulators In this section, we review the different definitions of the Z2 invariant for time reversal invariant free (or weak-interacting) fermionic topological insulators. For 2d and 3d time reversal invariant topological quantum systems, Z2 invariants characterizing topological insulators were introduced in [44, 32, 33, 55]. We will first review their definitions in this section. There are actually several natural definitions for these, using Pfaffians, polarizations, determinant line bundle and holonomy etc. Besides giving these definitions, we explain how they are equivalent. Physically, the Z2 invariant can be interpreted as the parity of so-called Majo- rana zero modes. In particular calling the invariant ν, for a 2d quantum spin Hall system, ν ≡ 0 (respectively ν ≡ 1) when there exist even (respectively odd) pairs of helical edge states. Note that one helical edge state consists of two chiral edge states. Similarly for 3d time reversal invariant topological insulators, ν ≡ 0 (respec- tively ν ≡ 1) when there exist even (respectively odd ) numbers of Dirac cones, i.e. conical singularities produced by surface states at the fixed points of time reversal symmetry. We will go through the details in several subsections. Section 3.1 is an introduc- tion to time reversal symmetry and Z2 topological insulators. We provide examples of two 2d and one 3d topological insulators. We then go on to define the 2d Kane– Mele invariant in its original version using Pfaffians and determinants, being careful to define the setup rigorously. We also discuss its 3d generalization in Sec. 3.2. Here 1630003-12 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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