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4.3. Summary It is shown in [47, 69] that for a 3d topological insulator, the Chern–Simons invari- ant and the Kane–Mele invariant are equivalent, i.e. υ = ν, since they are just the integral form and the discrete version of the mod 2 degree. Hence we call it the Z2 invariant of time reversal invariant topological insulators and view it as a topolog- ical index. Another proof of the equivalence of the Chern–Simons and Kane–Mele invariants was given in [31] based on the group of reduced topological phases, which will be discussed in the next section. Earlier, we discussed a mod 2 index theorem, proved in [35], connecting the analytical index ind2(D) (4.1) and the topological index (3.18) for a Dirac oper- ator with time reversal and chiral symmetries. Viewing this in the more general framework above, the reason why a mod 2 index theorem appears in Z2 topological insulators is the following. In an odd dimensional time reversal invariant fermionic system, the parity anomaly pops up as a global anomaly, which is really difficult to compute in general [2]. By translating the problem into a gauge theory, the global parity anomaly is equivalent to a gauge anomaly, which can be dealt with locally. In our case, the parity of the spectral flow of edge states characterizes the global property of the material, and the gauge theoretic WZW term provides a practical way to compute the parity anomaly locally. It is well known that an index theo- rem can be used to compute a global quantity by a local formula, for example, the odd index theorem computes the spectral flow by the odd Chern character as dis- cussed in Sec. 4.2.2. Therefore, the topological Z2 invariant can be interpreted as the mod 2 version of the odd index theorem, which computes the global parity anomaly locally. Let us recap the relations between different variants of the Z2 invariant dis- cussed above. Based on the non-abelian bosonization, the Z2 invariant describes the parity anomaly of Majorana zero modes in a time reversal invariant topological insulator. On the one hand, the analytical Z2 index is the parity of the spectral flow of edge states under an adiabatic evolution. The fermionic path integral of Majorana zero modes such as Dirac cones introduces the Pfaffian, while the path integral of the equivalent composite boson gives the square root of the determinant. After an adiabatic procedure, one can have a different way to bound two Majorana zero modes, so that the ratio of the effective action, i.e. the sign of the Pfaffian, would change accordingly. The Kane–Mele invariant keeps track of the mod 2 ver- sion of the change of the signs of Pfaffians, that is, the parity of the spectral flow of chiral edge states through the adiabatic evolution. For infinite dimensional Hilbert spaces, instead of the determinant (respectively Pfaffian) of matrices, we consider the determinant (respectively Pfaffian) line bundles of Dirac operators [29]. In the language of bundles, the Z2 invariant is obtained by comparing the orientations of the Pfaffian and the determinant line bundles over the fixed points of the time reversal symmetry. If we further consider the holonomy of the Pfaffian or the deter- minant line bundle, the quotient form of the Kane–Mele invariant is replaced by a difference form, since the holonomy of the Pfaffian or determinant line bundle is an 1630003-43 Notes on topological insulators 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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