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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann one can cobble together a section using both u1n and u2n on certain 1-dimensional domains, for instance in the case of a 1-dimensional Brillouin zone. In particular, let V be a small Θ stable path through a fixed point k0. Decom- poseV asV+⨿V− withT(V+)=V− andV+∩V− =k0.Nowwecanfixsections |u1n(k)⟩ and |u2n(k)⟩ independently on V + and V −, by continuing the states contin- uously to the fixed point k0 from both sides. By perturbation theory, we can choose these to match up if necessary after multiplying them with a local phase factor. lim u1n (k) = lim u2n (−k) and lim u2n (k) = lim u1n (−k). k→k0 −k→k0 k→k0 −k→k0 The reason for this is that after subtracting a constant term, the family is given by H(k) = x(k) · σ, with x(k0) = 0 and x(k) an odd function of k. This means that up to rotation, linearly near k0: H(k) = (k−k0)σz +O(k2), whence the claim follows by standard perturbation theory. Here we need that we assumed that the fixed points are isolated. These equations mean that one can glue two sections together over V , which we call |uI (k)⟩ and |uII (k)⟩ with |uI (k)⟩| + = |u1 (k)⟩, |uI (k)⟩| − = |u2 (k)⟩ and nnnVnnVn |uII (k)⟩| + = |u2 (k)⟩, |uII (k)⟩| − = |u1 (k)⟩. Likewise, in this case, we can glue nVnnVn together the functions χ1n(−k) and χ2n(k) at k0 to form one function χn(k). For these sections, we obtain that uII (−k) = eiχn(−k)ΘuI (k) and uI (−k) = −eiχn(k)ΘuII (k). (3.15) nnnn Strictly speaking, to follow the literature, we only need the 1d version for the base S1, with involution T (k) = −k with k ∈ [0, 2π] mod 2π. In this case we have fixed points 0, π and V + = [0, π], V − = [−π, 0]. We fix continuous normalized sections u1,2(k) on (0,π) and (−π,0) and extend them to V±. We relabel them n with Roman labels I and II as above. This is illustrated in Fig. 2. Then equation Ε uI uII u2 u2 11 uu −π uII Fig. 2. uI 0π Sections in the Brillouin zone. 1630003-20 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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