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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann bulk bands and denote the occupied bands by |k⟩ := Nn=1 |n, k⟩. Further assume that there are also N unoccupied bands and that there exists a band gap between the conduction (occupied) and valence (unoccupied) bands, and the Fermi level EF is assumed to lie in the band gap. If there is a physical material the Fermi level is determined by it, purely mathematically, without a given material, one can choose EF to be any energy in the band gap. Define the Fermi projection in ket-bra notations as N PF := |n,k⟩⟨n,k|=|k⟩⟨k|. (5.3) n=1 In addition, the Fermi level is always assumed to be zero, i.e. EF = 0, so that the occupied bands have negative eigenvalues. From now on we denote the Fermi projection by P ≡ PF for simplicity. For non-degenerate band structures, the spectral flattening trick can always be applied, which replaces the ith band Ei(k) (a continuous function over the Brillouin torus) by a constant function Ei(k) = Ei. Define the sign of the Dirac Hamiltonian by F(k) := H(k)|H(k)|−1, by which the band structure is reduced to a flattened two-band model, or in terms of the Fermi projection the flattened Hamiltonian F is written as F (k) = 1 − 2P. We use the notation F to indicate that it looks like the grading operator of a Fredholm module in K-homology satisfying F2=1, F∗=F. For the unitary class (type A), the flattened Hamiltonian F (k) defines a map from the Brillouin torus to the Grassmannian, aka. a nonlinear sigma model: F :Td →Gr(N,C2N)=U(2N)/(U(N)×U(N)) (5.4) since the occupied and empty bands each have N levels by assumption and we don’t distinguish energy levels among occupied or empty bands. Similarly, for the symplectic class (type AII), F defines another nonlinear sigma model, F :Td →Gr(N,R2N)=O(2N)/(O(N)×O(N)). (5.5) Without loss of generality, we assume that the empty bands have infinitely many levels. In other words, the 2N-dimensional total vector space is replaced by the infinite dimensional Hilbert space, then the classifying space BU(N) (respectively BO(N )) plays the role of the Grassmannian Gr(N, C2N ) (respectively Gr(N, R2N )). If N is big enough, the lower dimensional homotopy groups stabilize. Accordingly one can regard the target spaces the direct limits, BU = lim BU(N), BO = lim BO(N) −→ −→ NN which capture the stable (under increasing N ) features, which is what stable homo- topy theory is concerned with. 1630003-46 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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