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(3.15) holds and in this basis the matrix of Θ becomes wn(k)=(⟨usn(−k),Θutn(k)⟩)s,t=I,II = 0 e−iχn (−k) −iχn(k) −e ∈U(2). (3.16) 0 This matrix is skew-symmetric at fixed points k = −k and its Pfaffian there, which is just the upper right corner, is Pf(wn(k)) = −e−iχn(k). This type of argument also works for other regions V = V + ∪V − with T (V +) = V−,V0 :=V+∩V− ⊂Fix(T)s.t.allthepathcomponentsofV\V0 arecontractible and contained in either V+ or V−, i.e. there is no path from V+ to V− that does not go through a fixed point. 3.2.3. Definition In this subsection, we will briefly recall the definition of the Z2 invariant ν using time reversal polarization following [32]. Originally a Z2 invariant was introduced by Kane and Mele [44] in the study of the quantum spin Hall effect in graphene, whose geometry is that of the honeycomb model with spin-orbit interaction. A simplified account was given in Fu–Kane [32]. The Kane–Mele invariant will be interpreted as a mod 2 analytical index, so that it counts the parity of the spectral flow of edge states in Z2 topological insulators [41]. 1 Given the decomposition of B = Vn and restricting each Vn over an S , the above sections uI,II can be used to define an electric polarization in a cyclic adiabatic evolution. To this end, define the partial polarizations as the integral of the Berry connections, π −π where the Berry connections are locally defined as asn(k) = i⟨usn(k)|∂k|usn(k)⟩. Using the combinations of the partial polarizations, one further defines the charge polar- ization Pρ and the time reversal polarization Pθ, Pρ := PI +PII, Pθ := PI −PII. For time reversal invariant topological insulators, the charge polarization is always zero, and the time reversal polarization delivers the topological Z2 invariant. In particular, one computes the time reversal polarization for the codimension 1 submanifold {(k, 0) | k ∈ [−π, π]} ⊂ T2 in 2d. This corresponds to an edge (or edge states) of the effective Brillouin zone, which is a fundamental domain for the time reversal Z2 action. This embedded S1 has two fixed points, say Γ = {0, π} ⊂ [−π, π]. One computes, see e.g., [63, Chap. 4]: n 1 π P = dk(aI −aII)= 1 det(wn(π)) Pf(wn(π)) Pf(w (0)) Ps := asn(k)dk, s=I,II −2ln −π n n θ 2π n n 2πi 1630003-21 ln . Notes on topological insulators det(w (0)) 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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