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information will be contained in the second Chern class c2(B) ∈ H4(Td) or more generally the H4 of the base of the bundle. Indeed, for 4-dimensional time reversal invariant models the fundamental topological invariant is the second Chern number [61]. However, H4 is possibly non-zero only for manifolds of dimension greater or equal to 4, so it is not useful directly in 2d and 3d. This is the reason for the introduction of the Z2 invariant. For 3d, one can use dimensional reduction from 4d to 3d to understand how the Z2 invariant appears as the dimensional reduction of c2. The vanishing of c1(B), however, also implies that the first Chern class of its determinant line bundle vanishes, i.e. c1(Det B) = 0 ∈ H2(Td), since the cohomol- ogy of Td is torsion free. In addition, this determinant line bundle has a global section, which is one of the main ingredients of the Z2 invariant. The main key to understanding the Z2 invariant geometrically can be traced back to [32], which we now make precise. 3.2.2. Geometric setup One rough idea about how to obtain the invariant is given as follows. At any fixed point k0, k0 = −k0, Θ is a skew-symmetric (3.6) anti-linear morphism on the fiber over k. After choosing a basis, a natural object to study is the Pfaffian of the resulting skew-symmetric matrix. Since we can reduce to rank 2 bundles this is just given by the top right corner of a 2×2 matrix. The actual choice of basis plays a role, as the sign of Pfaffian changes if for instance the two basis vectors are permuted. The Kane–Mele invariant is a way to normalize and compare these choices of Pfaffians at all fixed points simultaneously. For this, one needs to make a coherent choice of normalization and show that the result is independent of the choice, which is what the invariant does. A key fact that is used is that the determinant line bundle has global sections. This can be seen in several different ways as we now discuss. To study the choice of basis, first observe that at any fixed point k0, k0 = −k0, Θ|φ⟩ and |φ⟩ are linearly independent by (3.5), so that we can choose Θ|u1n(k0)⟩ and Θ|u2n(k0)⟩, s.t. Θ|u1n(k0)⟩ = |u2n(k0)⟩ and Θ|u2n(k0)⟩ = −|u1n(k0)⟩ where the second equation follows from the first by Θ2 = −1. In contrast if |u1n(k)⟩ and |u2n(k)⟩ are states, i.e. local sections, defined on a subset V that is stable under T , i.e. T (V ) = V , and contains no fixed points, i.e. V ∩ F ix(T ) = ∅, then, since B is trivial over Td \F ix(T ) by the assumption that there are only degeneracies at the fixed points, and Θ commutes with H, we get that away from a fixed point u1n(−k) = eiχ1n(−k)Θu1n(k) u2n(−k) = eiχ2n(−k)Θu2n(k) (3.14) nn the relation Θ2 = −1 implies that eiχ1,2(−k) = −eiχ1,2(k). Together these two facts imply that the u1n do not extend as continuous sections to the fixed points in any T stable V that does contain fixed points. However, 1630003-19 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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