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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann Here wn is the transition matrix (3.16), After exponentiating both sides and using the identity eπi = −1, one obtains (−1)Pθ = Γi =0,π det(wn (Γi )) Pf(wn(Γi)) =: Γi =0,π sgn(Pf(wn(Γi))). It is important to point out that det(wn(k)) is a continuous choice of square root defined on the interval from 0 to π. This is possible by extending a choice of a branch of the square root at 0 along the interval [0, π] as in [32]. This is what gives the coherent choice of normalization mentioned above, which manifests itself as a choice of the sign of the Pfaffian. Since both the Pfaffian and the chosen branch are square roots of the determinant, they differ only by a sign ±1. There is a more highbrow explanation in terms of sections which will be presented subsequently. In terms of the time reversal polarization, the Kane–Mele invariant on S1 rela- tive to a choice of square root of the determinant is defined by ν ≡ Pθ (mod 2). In general, the Z2-valued Kane–Mele invariant ν is defined by ν detwn(Γi) Pf (wn (Γi )) equivalently given by (−1) = (3.17) Γi ∈Γ for the fixed points Γ of the time reversal symmetry. Here, again, one has to be careful to say how the branches of the square root of det are chosen. This is dictated by the embedding of S1s, which pass through the fixed points. In the 2d case these are chosen to be {(k,0)} and {(k,π)} in [32]. This is actually not quite enough as realized in [32] and explained in [33]. Namely one needs a branch of the square root defined on all of T2. Since c1(DetB) = 0 there is indeed a global section. One can then argue that this section has a square root [31, 26], and below Sec. 3.3.3. Notice that this is possible, but it is not possible to find a basis of global sections, since the bundle need not be trivial. Unlike the case for S1, for the 2d invariant, the result is actually invariant under such a choice of section, see [33, 31]. The 3d case is discussed in the next section. 3.2.4. Spectral flow In the appendix of [32], Fu and Kane showed that the 2d Kane–Mele invariant is 1 ν ≡ 2π a − f mod 2 (3.18) ∂EBZ EBZ where a and f are the Berry connection and curvature, EBZ is the 2d effective Brillouin torus with open cylindrical geometry and its boundary ∂EBZ consists of two circles. The equivalence can be proved by expressing the partial polarizations by integrals of the Berry connection and curvature. One has to calculate modulo 1630003-22 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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