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To eliminate the ambiguity in choosing the integers from the complex argument of the logarithmic function, we modulo both sides by 2π, πh ≡ i ln Pf(wn(π)) (mod 2π). Pf (wn (0)) It is easy to get an equality by exponentiating as before, (−1)h = Pf(wn(0)) . (3.21) Pf(wn(π)) The Pfaffian of wn is well-defined only at the fixed points k = 0,π, but the determinant of wn is defined everywhere for k ∈ [0, π]. Let us start with the equation satisfied by the determinant, 1 π 1 d[lndet(wn(k))] = 2 ln det(wn(π)) det(w (0)) det(wn(π)) = ln . Notes on topological insulators 2 For any skew-symmetric matrix A, the sign of the Pfaffian satisfies, 0 n det(wn (0)) Pf2A = det A, sgn(PfA) = √det A/PfA. Then the above can be written in the Pfaffian and its sign, det(wn(π)) Pf(wn(π)) sgn(Pf(wn(π))) ln det(wn(0)) = ln Pf(wn(0)) + ln sgn(Pf(wn(0))) . Change the determinant into a trace, we have 1π 1π 1π d[ln det(w (k))] = d [tr ln w (k)] = dk tr(w−1∂ w ). 2n2n2nkn 000 Therefore, we have Pf(w (π)) 1 π Pf(wn(0)) n 20 nkn ln n +ln sgn(Pf(w (α)))= dktr(w−1∂ w ). α=0,π In order to compare it with the holonomy, we exponentiate it, sgn(Pf(wn(Γi))) Then the above is equivalent to, (−1)−h = (−1)n+ν. In other words, the Z2 invariant ν is equivalently defined by ν≡n−h (mod2). Pf(wn(π)) n(w ) Pf(wn(0)) = (−1) n where the half winding number n is defined by Γi =0,π 1π n(w ) := n2πi0 nkn (3.22) (3.23) dk tr(w−1∂ w ). In contrast to the Kane–Mele invariant being a quotient in the previous subsection, this difference form is another way to compare the square root of the determinant (as a half winding number) with the Pfaffian (as a holonomy). 1630003-25 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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