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Furthermore, given a map from a compact (2k + 1)-dimensional spin manifold M to the unitary group g : M → U(n), its homotopy class is an element in the odd K-group, i.e. [g] ∈ K−1(M). The odd Chern character of g is defined by [37] ∞ k! ch(g) := (−1)k tr[(g−1dg)2k+1] (4.18) which is a closed form of odd degree. Or equivalently, ch(g) is the relative Chern– Simons form cs(d,g−1dg) by Taylor expansion. As a classical example, the degree ofg:S2k+1 →U(n)isgivenby i k deg(g) = − ch(g). Replacing connections by Dirac operators, the role of the relative Chern–Simons form cs(d,g−1dg) is played by the spectral flow sf(D,g−1Dg) [37]. where Dt =(1−t)D+tg−1Dg, D ̇t =g−1[D,g]. The Dirac operator D defines a Fredholm module in K-homology, and the spectral flow computes the Fredholm index of the Toeplitz operator PgP by the pairing between K-homology and K-theory [37], index(PgP) = ⟨[D],[g]⟩ = −sf(D,g−1Dg) (4.20) where P := (1 + D|D|−1)/2 is the spectral projection. Baum and Douglas [15] first noticed the odd Toeplitz index theorem, which is an identity connecting the analytical index and topological index, M where Aˆ is the A-roof genus, for a generalized odd index theorem for manifolds with boundary see [24]. In particular, we have Aˆ(T3) = 1 since Aˆ is a multiplicative genus and Aˆ(Sk) = 1 for spheres. Hence the degree of g can be computed as the spectral flow on the 3d Brillouin torus, k=0 (2k + 1)! s f ( D , g D g ) = √ π t r ( D ̇ t e t ) d t ( 4 . 1 9 ) 0 Notes on topological insulators 2π S2k+1 The analytic spectral flow of a Dirac operator D on M can be computed as −1 11 −D2 s f ( D , g − 1 D g ) = Aˆ ( M ) ∧ c h ( g ) ( 4 . 2 1 ) i 2 sf (D, g−1Dg) = − ch(g) = deg g. (4.22) 2π T3 Finally, we apply the odd index theorem (4.22) to the Dirac Hamiltonian H of a 3d topological insulator and the transition matrix w (3.16), then the Chern–Simons 1630003-39 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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