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with boundary the spectral flow of a family of Dirac operators can also be computed by the Maslov triple index, the interested reader may consult for example [50]. In the simplest symplectic vector space (R2,ω) with the standard symplectic form ω, L ⊂ R2 is a Lagrangian subspace, or simply Lagrangian, of (R2,ω) if L = L⊥, which is equivalent to L being 1-dimensional in R2. Denote the 1-dimensional Lagrangian Grassmannian Gr(1,2) in (R2,ω) by Λ(1), and each element in Λ(1) is represented by a line L(θ) with slope θ. The square function Sq : Λ(1) → S1; L(θ) → e2iθ induces a diffeomorphism Λ(1) ≃ U(1)/O(1) ≃ RP1 and the fundamental group of Λ(1) is π1(Λ(1)) = π1(S1) = Z. In (R2,ω), the geometric Maslov index is defined as the intersection number of a line L(θ) ∈ Λ(1) with a reference line usually fixed as R = L(0). Remem- ber that in the 2d quantum spin Hall system, we model the chiral edge states by 1d Lagrangians, and we are interested in their intersections with the Fermi level. One version of analytical Maslov index is defined by the eta invariant of a real self-adjoint Dirac operator D(L1, L2) where L1, L2 ∈ Λ(1). Another version of ana- lytical Maslov index is defined by the spectral flow of a family of Dirac operators {D(L1(t),L2(t)),a ≤ t ≤ b}. In other words, the geometric Maslov index can be interpreted as the spectral flow of a family of Dirac operators, which is intimately related to the eta invariant. We now show how to compute the analytical Maslov indices explicitly in (R2,ω). Denote the unit interval by I = [0,1], let W1,2(I;L1,L2) be the Sobolev space as the completion of the space of paths connecting two Lagrangians L1,L2, i.e. φ ∈ C1(I, R2) s.t. φ(0) ∈ L1, φ(1) ∈ L2. The Dirac operator D(L1, L2) defined by D(L1,L2) : W1,2(I;L1,L2) → L2(I,R2); φ → −idφ (4.2) dt is a Fredholm operator and its kernel is given by constant functions Ker(D(L1, L2)) = {φ(t) = constant point ∈ L1 ∩ L2}. Notes on topological insulators When Re(s) ≫ 0, the η-function of a self-adjoint elliptic operator D is defined by 11 η(D,s)= λs − (−λ)s (4.3) λ>0 λ<0 which admits a meromorphic extension to the complex plane and is regular at s = 0. One defines the eta invariant as η(D) := η(D,0), and the reduced eta invariant η ̃(D) as η ̃(D) := η(D) + dim(ker D) . (4.4) 2 Roughly, the eta invariant is the number of positive eigenvalues minus the number of negative eigenvalues. In symplectic geometry, one defines the analytical Maslov 1630003-33 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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