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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann In particular, for the 2d case (i.e. k = 2) π2(H2) = Z, which reflects the existence of the TKNN integers (or the first Chern number) as the Z-invariant. As mentioned previously, from [68], we know that for a family of Hamiltonians H ∈ H, its eigenvalue degeneracy is generically codimension three, so in 3d a S2 can be used to enclose the monopole and the first Chern number of the restriction of the Bloch bundle provides a topological invariant, see Sec. 2.3. This corresponds to the fact π2(H2) = Z. If one includes time reversal symmetry, two more spaces become of interest. The first is HKr, which is the subspace of Hermitian matrices allowing possible pair degeneracies, that is Eigenvalues 1 and 2 or Eigenvalues 3 and 4 may be degenerate and so forth. This is the space for families of Hamiltonians which have at most Kramers degeneracy. The second is the subspace HΘ HΘ ={H|ΘHΘ−1 =H} (3.29) where now Θ is the given action of time reversal on the Hilbert space, that is an anti-unitary operator with Θ2 = −1. To not overburden the notation we omitted the index k, but we will always assume that the dimension of the family of Hamiltonians is fixed. Moreover, for time reversal invariant systems the interesting case is when the Bloch bundle has even rank, so we will assume that k = 2n. In this case, independent of the particular choice Θ, each element in HΘ can be represented by a non-degenerate quaternionic n × n Hermitian matrix, and the space HΘ is homotopic to Sp(n)/Sp(1)n, see e.g., [12]. The relation is that the family of Hamiltonians over the Brillouin torus can be viewed as a map H : Td → H. If we add the time reversal symmetry, we additionally have the two Z2 actions T on Td and conjugation by Θ on H. The map H then becomes a Z2 equivariant map. The condition that there are only the degeneracies forced by Kramers degeneracy implies that in fact H : Td → HKr. Since this map is still equivariant it is determined by the restriction H|EBZ : EBZ → HKr. Said differently, under the time reversal symmetry, we only consider the half Brillouin torus giving a fundamental domain for the Z2 action, i.e. the EBZ, since the other half can be easily recovered by reflection. Furthermore, at the fixed points Γi ∈ Γ the Hamiltonians are invariant, so that H|Γ → HΘ ⊂ H. The lower homotopy groups of time reversal invariant Dirac Hamiltonians are known, see [12]: π1(HΘn ) = π2(HΘn ) = π3(HΘn ) = 0, π4(HΘn ) ≃ Zn−1. Analogously to the above, for a generic time reversal invariant family of Dirac Hamiltonians H ∈ HΘ2 , its eigenvalue degeneracy has codimension five and Kramers doublets are distinguishable in 4d families, e.g., by enclosing the degeneracies by S4. In other words, the homotopy classes of maps [S4, HΘ2 ] = π4(HΘ2 ) = Z are labeled by Z, which is just the second Chern number c2 of the Bloch bundle. Thus c2 manifests itself as a fundamental invariant for time reversal invariant models, which follows 1630003-30 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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