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and are conical one has Dirac cones. The number of Dirac cones determines whether the system is a topological insulator, that is, if a material has odd number of Dirac cones, then it is a stable 3d topological insulator. For example, the effective Dirac Hamiltonian for Bi2Se3 with spin orbit coupling has a single Dirac cone [74]. The Hamiltonian is H(k) = A1kxσx ⊗sx +A1kyσx ⊗sy +A2kzσx ⊗sz +M(k)σz ⊗s0 +ε0(k)I4 where I4 is the 4 × 4 identity matrix, M ( k ) = M − B 1 k z2 − B 2 ( k x2 + k y2 ) ε0(k) = C + D1kz2 + D2(kx2 + ky2) (3.12) and the Ai, Bi, C, Di and M are model parameters. 3.2. Kane–Mele invariant In this section, we carefully present the relevant structures that the time reversal symmetry induces on a Bloch bundle and then use these structures to define the Kane–Mele invariant in 2d and 3d, following their original construction and finish with a holonomy interpretation of the invariant. We will also discuss a coordinate free bundle theoretic description of these constructions based on canonical orienta- tions in the following section. 3.2.1. Kramers degeneracy In this subsection, we discuss the geometric setup for time reversal invariant topo- logical insulators. We are following physical notations, this means that we use sec- tions. These are allowed to be non-zero only locally, or in other words locally defined. So we have to stress that in order to get an global object, sometimes extra work is needed to glue local sections together. Consider a time reversal invariant Dirac Hamiltonian H(k), k ∈ Td, whose eigenvalue equation is H(k)un(k) = En(k)un(k). For each physical state |un(k)⟩ = |n, k⟩, Θ|n, k⟩ gives another state with the same energy, and such doublet is called a Kramers pair. At a time reversal invariant point k = T (k) = −k ∈ Td time reversal symmetry forces degeneracy or a level crossing. That is both |un(k)⟩ and Θ|un(k)⟩ are states with momentum k = T (k) = −k and the same energy. They are linearly independent, due to (3.5). This is called Kramers degeneracy. Starting with a lattice model, in the reciprocal lattice the time reversal invariant points are given by −Γi=Γi+ni·K, i.e.Γi=1ni·K, 2 where K is the reciprocal lattice vector and the coordinates of ni are 0 or 1. For example, in a 2d square lattice with K = (2π,2π), we have four time reversal invariant points Γ = {(0,0),(0,π),(π,0),(π,π)}. So, for instance, the Brillouin Notes on topological insulators 1630003-17 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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