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Acn) that encodes the information about non-trivial extensions of free Dirac Hamil- tonians of different dimensions in the periodic table. Furthermore, the Clifford modules and the cokernels inherit the Bott periodicity: Ak+8 ≃ Ak, Ack+2 ≃ Ack. In fact, the cokernels have the same form as the homotopy groups of the classifying spaces [5], An ≃ πn(BO), Acn ≃ πn(BU). For instance, if we extend Cl0,1(R) generated by T in the quantum spin Hall effect, i : Cl0,1(R) ≃ C → Cl0,2(R) ≃ H then A1 ≃ Z2 gives the 2d Z2 invariant. One step further, if we extend Cl0,2 generated by T and QT for 3d topological insulators, i : Cl0,2(R) ≃ H → Cl0,3(R) ≃ H ⊕ H then A2 ≃ Z2 shows that there exists another Z2 invariant in 3d. It is easy to promote vector spaces to vector bundles and get the K-theory of Clifford bundles, that is, to pass from the Clifford algebra classification to the K- theory classification of topological insulators. For any pseudo-Riemannian vector bundle E → X over a compact Hausdorff space X, define an associated Clifford bundle Cl(E) → X as usual, and a representation gives a Z2-graded vector bundle M(E) → X. The Clifford extension problem arises naturally when we add a one- dimensional trivial bundle T1 → X by the Whitney sum, and the cokernel bundle is defined as A(E) := coker(i∗ : M(E ⊕ T1) → M(E)). By the Clifford grading M(E) = M0 ⊕ M1 and its Euler class is defined as the formal difference χE(M) := [M0] − [M1] which induces a map from the cokernel bundle to the reduced KO-group, + χE : A(E) → KO(B(E), S(E)) = KO(E ) where the right-hand side is the Thom construction, B(E) and S(E) are the ball and sphere bundles of E, E+ is the one-point compactification. In particular, if the vector bundle E is of rank n, and assume the base space is contractible X ∼ pt, then A(E) is reduced to An and we get the isomorphism An ≃ KO(Sn) = πn(BO) More details can be found in [5]. On the level of Clifford modules, the Clifford extension problem is described by the triple (E,F;σ), where E,F are elements in M(Cl0,n+1(R)) and σ is a lin- ear orthogonal map identifying E|Cl0,n = F|Cl0,n as restricted representations of 1630003-49 Notes on topological insulators 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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