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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann exponentiated eta invariant with a multiple of iπ [19]. If we model the edge states by Lagrangian sub-manifolds in the symplectic setting, then the spectral flow can be computed by the Maslov index, that is, the Lagrangian intersection number of the chiral edge states with the Fermi level. On the other hand, the topological Z2 index is the topological Wess–Zumino– Witten (WZW) term modulo two, since the bosonization process delivers the WZW term as the action functional of chiral edge currents [73]. The bulk-edge correspon- dence in this context is given by the Chern–Simons/WZW duality, and the WZW term is just the topological index of odd Chern characters of gauge transformations. In our case, the time reversal symmetry induces a specific gauge transform, so the Z2 invariant is the mod 2 index of the odd Chern character of this special unitary in the odd K-group. Hence the mod 2 version of the odd index theorem computes the spectral flow of the edge Dirac Hamiltonian, so the topological Z2 invariant is interpreted as the Witten index Tr(−1)F. If we use the Feynman propagator instead of gauge transformations, the WZW term can be calculated based on the method of Green’s functions. The homotopy theory of Dirac Hamiltonians tells us that the 2nd Chern number, i.e. the integral of the second Chern character is the fundamental invariant characterizing time reversal invariant models. In addition, the homotopy theory on the effective Hamiltonians gives us an interesting explana- tion of the Z2 invariant. According to the dimensional ladder in anomalies [56], the 3d parity anomaly is the descendant of the 4d chiral anomaly, so the WZW term (its mod 2 version is the parity anomaly) is naturally connected to the second Chern character (which computes the chiral anomaly). In other words, the topological Z2 invariant can be derived from the second Chern number from the relation between anomalies. Finally an analogy of the global SU(2) anomaly of the WZW model explains that it is naturally Z2-valued, so we identify the topological Z2 index as the mod 2 WZW term, and it is called the parity anomaly instead of the SU(2) anomaly in condensed matter physics. 5. K-Theoretic Classification In this section, we will discuss the K-theoretic classification of non-interacting (or weak interacting) Dirac Hamiltonians of topological insulators. More precisely, the Z2-valued topological invariants of topological insulators and Bogoliubov–de Gennes (BdG) superconductors fit into a periodic table resembling the Bott period- icity as in topological K-theory. First, a Clifford algebra classification for these was established in [62] and the K-theory classification followed as the Clifford extension problem [51]. A finer classification of topological phases based on KR-theory and twisted equivariant K-theory was proposed in [31]. For time reversal invariant topo- logical insulators, KH-theory was also applied to the classification problem in [26]. KR-theory, introduced by Atiyah [4], is the K-theory of Real vector bundles over an involutive space. By contrast, KH-theory is the K-theory of Quaternionic vector bundles over an involutive space. They are isomorphic to each other by a Fourier 1630003-44 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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