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we first consider the general structure of Bloch bundles with time reversal Z2 sym- metry. The next subsection Sec. 3.3 then gives a geometric construction of the Z2 invariant using bundle theoretic means. While Sec. 3.4 reviews the homotopy the- oretic characterization of the Z2 invariant that leads over to the considerations of Sec. 4. 3.1. Background and examples We first recall the time reversal symmetry, which introduces Kramers degeneracy and changes the effective topology and geometry of the game. Next, we review the effective Hamiltonians of Z2 topological insulators, which can be used to define elements in K-homology in the future. 3.1.1. Time reversal symmetry A physical system has time reversal symmetry if it is invariant under the time reversaltransformationT :t→−t.Inquantumsystemsthetimereversalsymmetry is represented by a time reversal operator Θ. By general theory [71], Θ is necessarily anti-unitary and can be realized as the product of a unitary operator and the complex conjugation operator. Again by general theory, in momentum space (or the Brillouin zone) T acts as k → −k. A time reversal invariant model is required to have [H(r),Θ] = 0, or in the momentum representation ΘH(k)Θ−1 = H(−k). (3.1) For example, in a two-band system the time reversal operator Θ is defined by Θ = iσyK (3.2) where σy is the imaginary Pauli matrix and K is the complex conjugation. Θ is indeed an anti-unitary operator, i.e. for physical states ψ and φ, ⟨ Θ ψ , Θ φ ⟩ = ⟨ φ , ψ ⟩ , Θ ( a ψ + b φ ) = a ̄ Θ ψ + ̄b Θ φ , a , b ∈ C . ( 3 . 3 ) For a spin-1 particle, such as an electron, it also has the property: 2 Θ2 = −1. (3.4) This results in the so-called Kramers degeneracy, which is the fact that all energy levels are doubly degenerate in a time reversal invariant electronic system with an odd number of electrons. In fact, φ and Θ(φ) are orthogonal: ⟨Θφ, φ⟩ = ⟨Θφ, ΘΘφ⟩ = −⟨Θφ, φ⟩ = 0. (3.5) In general, Θ is skew-symmetric in the sense that ⟨Θψ, φ⟩ = ⟨Θφ, ΘΘψ⟩ = −⟨Θφ, ψ⟩. (3.6) The Dirac Hamiltonian H(k) = kxσx + kyσy + (M − Bk2)σz is not time rever- sal invariant. In contrast, the models for quantum spin Hall effect with spin-orbit coupling given in the next section are time reversal invariant systems. Notes on topological insulators 1630003-13 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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