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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann 3.3. Topological band theory With the additional structure given by the time reversal symmetry, the Hilbert bundle modelling the band structure becomes a Quaternionic vector bundle. Specif- ically, the time reversal Z2 symmetry introduces an involution on the Brillouin torus and a Quaternionic structure on the physical Hilbert space, that is the sections of the Bloch bundle. Over the time reversal invariant points, the quaternionic struc- ture gives rise to a symplectic structure of the restricted Bloch bundle. 3.3.1. Quaternionic structure In the bundle theoretic framework, one obtains the Hilbert space as the space of sections of the Bloch bundle H = Γ(B). Since the bundle B splits as Vn, so does n the Hilbert space as n Hn, with Hn = Γ(Vn). With the extra structure of the time reversal operator Θ, the complex Hilbert space H(C) = H can be viewed as a quaternionic Hilbert space H(H), by setting j = Θ, with j2 = Θ2 = −1. The action of H = spanR⟨1,i,j,k⟩ = spanC⟨1,j⟩ is given by i, i.e. multiplication by i, j acting as Θ, which is anti-unitary and hence anti-commutes with i, and k = ij = iΘ. Θ respects the decomposition of H, so that each Hn becomes a H vector space Hn(H) and H(H) = Hn(H). The fibers n of Vn over a fixed k are 2-dimensional complex vector sub-spaces. Θ in general permutes these sub-spaces, except over the fixed points. At these points the fiber becomes a quaternionic vector space. That is B|Γ is a quaternionic bundle and each fiber VnΓi ∼= H, that is, a 2d C vector space becomes a 1d H vector space. In particular HΓ = Γ(B,Γ) is an H invariant subspace. Over Γ, if |un⟩ is a (local) basis over H then |un⟩ and |Θun⟩ is a basis over C. These are usually put into a so-called Kramers pair |Ψn⟩ := (|un⟩, Θ|un⟩) ∈ HΓ(H) ⊂ Hn(H) then picking (|un⟩,Θ|un⟩) as a basis, we see that 01 Θ·Ψn=JΨn, J=iσy= −1 0 The relation to the Pfaffian is as follows: define a bilinear form on the complex where Ψn is viewed as a 2-vector. Hilbert space H by ωΘ(φ, ψ) = ⟨Θφ | ψ⟩. (3.24) This form becomes a symplectic form for Vn over the time reversal invariant points; ωΘ ∈ ∧2(Vn)∗. Thus Vn|Γ → Γ is a symplectic vector bundle. The associated almost complex structure is Jn = −iσy. Hence the Bloch bundle restricted to the fixed points Vn|Γ → Γ carries an associated almost complex structure JΘ = Jn. We n have that Θ = KJ. 1630003-26 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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