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invariant as the product of signs of the invariant section ν = Γi ∈Γ sgn (s(Γi)) = s(Γi) Pf(Γi) Notes on topological insulators where Pf : Γ → DetΘBΓ is a global section representing the canonical orientation determined by the Pfaffian form. That is, the sign is positive/negative if s is in the positive/negative half line determined by the orientation. Abusing notation, one can set s = det since it is Det Θ invariant, and rediscover the Kane–Mele invariant. In other words, the Z2 invariant can be defined equivalently by comparing the orientations of the restricted determinant line bundle and the Pfaffian line bundle over the fixed points. In [26], a so-called FKMM-invariant was constructed almost along the same lines as above. The restricted determinant line bundle DetB|Γ is trivial since Γ is a finite set. Furthermore there is a canonical isomorphism d e t Γ : D e t B | Γ ∼= Γ × C . This section is related to the above equivariant section by √det = det−1. The Real determinant line bundle (Det B, Det Θ) has an associated equivariant trivialization hdet : DetB ∼= Td ×C, and its restriction hdet|Γ gives another trivialization of Det B|Γ. The composition hdet|Γ ◦det−1 :Γ×C→Γ×C Γ defines a map ωB :Γ→Z2, s.t.(hdet|Γ◦det−1)(x,λ)=(x,ωB(x)λ). (3.26) Then the FKMM-invariant is defined as the equivariant homotopy class of ωB up to equivariant gauge transformations κ(B) := [ωB] ∈ [Γ,U(1)]Z2/[Td,U(1)]Z2. (3.27) In general, κ defines an injective map from Quaternionic vector bundles to relative √ Γi ∈Γ equivariant Borel cohomology classes over an involutive space (X,ς), κ:Vect2m(X,ς)→H2 (X,Xς;Z(1)). (3.28) Q Z2 See [26, Sec. 4] for more details about the injectivity, the equivalence between the FKMM-invariant and the Kane–Mele invariant was also proved in that section. 3.4. Homotopy theory The study of homotopy theory of time reversal invariant Hamiltonians can be traced back to the 80’s [12, 13]. Denote the space of non-degenerate Hamiltonians, i.e. non- degenerate k × k Hermitian matrices by Hk. It is homotopic to U(k)/U(1)k. The lower homotopy groups were computed in [13], π1(Hk) = 0, π2(Hk) = Zk−1. 1630003-29 Γ Γ 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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