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For electrons in a crystal, the electronic states are described by Bloch wave functions ψ(r) = eik·ru(r), u(r + R) = u(r), ∀ R ∈ Γ. (2.4) In the following, we use the momentum representation. The quasi-periodic eigen- states un(k) are by definition solutions to the eigenvalue equation, H(k)un(k) = En(k)un(k) where En(k) is called the nth band function. We assume that the En(k) are non- degenerate everywhere (i.e. no band crossing). In general, the energy spectrum {En(k), n ≥ 1} is referred to as the band structure of the model. Mathematically, the un(k) are sections of a Hilbert bundle E → B whose fiber over k ∈ B is given by the un(k) with fixed k. Here one uses L2 sections and the compact-open topology; see e.g., [31] for a detailed study. All the sections form a Hilbert space, which will be called H. The sections un(k) with fixed n form the sub-Hilbert space Hn and H = Hn. n In the physical literature sections are called physical states and are written as |un(k)⟩, or |n, k⟩ to emphasize the quantum numbers. Furthermore they are always assumed to be normalized so that ⟨n, k | n, k⟩ = 1. There are two types of bands: the occupied bands, literally occupied by the electrons, and the empty bands. The valence band is the top one among the occupied bands and the conduction band is the bottom one among the empty bands. The band gap is defined as the energy gap between the valence and conduction bands. The Fermi energy EF is defined such that in the ground state all single-particle levels with smaller energy are occupied and there is no occupied level above this energy. Loosely, it is the top of the collection of occupied electron energy levels. It coincides with the chemical potential at absolute zero temperature [3], which is the most precise definition. What the position of this energy inside the band structure is, depends on the specific material. For insulators EF lies in a band gap, which we will now assume. The construction of the bundles is valid for generalized Brillouin zones such as Sd or Td, but in the rest of this subsection we only use T2 for concreteness, which is the Brillouin zone in the 2d square lattice quantum Hall model. Let us consider the rank one (aka. line) bundle for the nth occupied band over the Brillouin torus. Ln → T2. (2.5) This is the eigenbundle to the nth energy band En, which is assumed to be below the Fermi energy. Notice that since the eigenvalues of the Hamiltonians are real, there is an order on them. Its sections are the nth eigenstates |un(k)⟩ : T2 → Ln; k → un(k). Assume that there exist N occupied bands, then the Bloch bundle is defined to be the rank N bundle given by the Whitney sum, B = Ln → T2. 1630003-7 N n=1 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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