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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann The Bloch bundle is a finite rank Hilbert bundle, whose space of sections is the physical Hilbert space. The sections themselves are Bloch wave functions. The Berry connection is then defined by a covariant derivative on the Bloch bundle [31], whose local form on the component Ln is given by an(k) = i⟨n, k|d|n, k⟩ (2.6) so the Berry curvature on Ln has a local expression fn(k) = id⟨n, k| ∧ d|n, k⟩. (2.7) The integral of the Berry curvature over the Brillouin torus gives the first Chern number, ∂an ∂an cn1=fn=dk2 y−x. (2.8) T2 T2 ∂kx ∂ky For the Bloch bundle B → T2, the total first Chern number is then the sum N c1= cn1∈Z. n=1 The generalized Bloch–Hamiltonian for the quantum Hall effect (QHE) in a magnetic field in the Landau gauge given by (0,eBx) is [66]: ˆ 1∂2 H(k1,k2)=2m −i∂x+k1 1∂ 2 +2m −i∂y+k2−eBx +U(x,y) U (x, y) = U1 cos(2πx/a) + U2 cos(2πy/b) (2.9) with the potential and boundary conditions uk1k2 (x + qa, y) exp(−2πipy/b) = uk1k2 (x, y + b) = uk1,k2 at rational flux p/q, see [66] for details. In particular the action of Z2 is restricted to a bigger sub-lattice, such that the non-commutativity is lifted. As was shown in [66] and in the following interpretations [14, 16] the quantization of the Hall conductance can be explained as the Chern class of a line bundle as discussed above. Fixing such a line bundle, for instance with the first Chern number n = 1 is frequently called a quantum Hall state. If this type of state is put next to an insulator, which has first Chern number n = 0 then at the edge there are localized chiral edge modes [41]. This produces so called helical edge states as these states propagate towards the edge, where they get reflected, since there is a different topology in the neighboring bulks, but cannot penetrate deep into their own bulk, again because of the topological obstruction, see [41, Fig. 2]. 1630003-8 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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