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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann a period 8, KHi(X) ≃ KHi+8(X). The relation between the KH-groups and the KR-groups is given by KHi(X) ≃ KRi−4(X). Since the Bloch bundle (B, Θ) → (Td, T ) has a Quaternionic structure (Θ2 = −1) rather than a Real structure, it is better to use KH(Td) for time reversal invariant topological insulators. However, the above relation tells us that if we shift the KR- groups by 4, then the KR-classification [31] agrees with the KH-classification [26] for the classification problem of Bloch bundles over the Brillouin torus for time reversal invariant models. 3.3.3. Determinant line bundle In this subsection, we first give another equivalent definition of the Kane–Mele invariant by looking at the determinant line bundle following [31]. Then we discuss the closely related FKMM-invariant in equivariant cohomology following [26]. The calculations of Sec. 3.2.6 yield the equivalence to the classical definition of the Z2 invariant using integrals. For the rank 2N Bloch bundle (B, Θ) → Td consider its determinant line bundle, which is given by: DetB := ∧2NB → Td. Taking the 2Nth exterior power DetΘ of Θ, we obtain an anti-linear morphism that squares to +1. This yields a Real bundle (Det B, Det Θ), and when restricted to the fixed point set Γ a real structure (Det B|Γ, Det Θ|Γ). The Det Θ invariants of Det B|Γ form a real sub-bundle, which is denoted by DetΘBΓ. This real bundle has an orientation given by the first component of the Bloch states. This means that if the |un⟩ are a quaternionic basis, then |u1⟩∧Θ|u1⟩∧···∧ |uN ⟩∧Θ|uN ⟩ is a canonical orientation. It is Det Θ invariant, since Det Θ(φ∧Θφ) = Θφ∧Θ2φ = −Θφ∧φ = φ∧Θφ. It is canonical since the space of quaternionic bases is connected. Alternatively, the symplectic structure ωΘ defines an anti-linear 2N form — aka. the Pfaffian form — which is ω∧N Pf(ωΘ):= Θ = ωΘ ∧···∧ωΘ N! . N! It restricts to an R-linear form on the real bundle DetΘBΓ yielding an orientation. This follows from ωΘ(Θ u, Θv) = wΘ(u, v). As mentioned above, c1(DetB) = 0 and hence there is a nowhere vanishing section. We can make this section real, i.e. land in the DetΘ invariant part by averaging, see e.g., [31] for details. We will denote this section by s. By comparing s(Γi) with the canonical orientation of the real line DetΘBΓ, one defines the Z2 1630003-28 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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