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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann 3.1.2. Effective Brillouin zone The association k → H(k) is formally a map H : Td → Hermn, where Hermn is the space of Hermitian n × n matrices. If we take into account the action of T on Td, then to specify a map with time reversal symmetry it is enough to know the restriction of H onto a fundamental domain of the T action, by (3.1). Such a choice of fundamental domain will be called Effective Brillouin Zone (EBZ). In the 2d-case such an effective Brillouin zone is given by a cylinder C = S1 × I ⊂ T2 and in 3d by T2 × I ⊂ T3, see Fig. 1. If one considers the boundary of this EBZ, then it still has a non-trivial action of T and we can consider a fundamental domain of the boundary of the EBZ, which we call Rd−1. In physics parlance, this region characterizes the edge states using a unique coordinate. For example, R1 = [0,π] in the 2d effective Brillouin zone [0,π] × [−π,π] and R2 is the square [0,π] × [0,π] in the 3d case, see Fig. 1. Although the usual Brillouin zone is Td, the spheres Sd play a special role. First they can be viewed as local invariants as in Simon’s interpretation of the TKNN integers [65] or generally as enclosing monopoles or higher codimension degeneracy loci. For instance, from [68], we know that for a family of Dirac Hamiltonians H ∈ H, its eigenvalue degeneracy is generically of codimension three, so in 3d for a generic family each point k corresponding to a degenerate eigenvalue, aka. monopole, is enclosed by an S2. Secondly, the spheres Sd occur through effective maps as in the Moore–Balents description below Sec. 3.4, or the more complicated arguments presented in Sec. 5.2. Finally, they appear in the stable splitting, see Sec. 5.4. In particular, they arise through pull-back via the collapse map q : Td → Sd, see loc. cit. Thus, we will look at maps H : X → Hermn where X can be Td,Sd, or Rd−1. Depending on the situation, the target may be a subspace of all Hermitian matrices, see Sec. 3.4. Fig. 1. (Color online) Effective Brillouin zone in 2d and 3d. 1630003-14 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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