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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann Using the so-called cyclotron frequency ω = eB/m, the Hamiltonian is written as p2 1 k2 H= x + mω2 x+ y (2.2) energy spectrum is given by En= 1+n ω, n≥0. 2 The set of wave functions (eigenspace) with energy corresponding to an integer n is called the nth Landau level. Denoting the harmonic oscillator eigenstates by ψn, the wave function of the free electron in our model then is expressed as φn(x,ky)=eikyyψn(x+x0), x0 = ky . (2.3) mω This state (2.3) is the first example of an edge state, which by definition is the eigenfunction satisfying the eigenvalue equation of the edge Hamiltonian (2.2). In general, given a bulk Hamiltonian Hb(x, y) which has translational invariance along one direction — say the y-direction — an edge Hamiltonian is obtained from the bulk Hamiltonian by a partial Fourier transform. The momenta ky in y-direction can then be treated as a parameter, so that the edge Hamiltonian becomes a one- parameter family He(x,ky) as do the energy levels and the eigenfunctions. That is, we have the following family of equations: He(x,ky)φn(x,ky) = εn(ky)φn(x,ky) parametrized by ky. In topological band theory, the band structure, i.e. the Landau levels, defines a complex vector bundle over the momentum space, more details will be given in the next subsection. In the simplest case, the band structure of a free Hamiltonian gives rise to a trivial vector bundle. 2.2. Bloch bundle Thouless, Kohmoto, Nightingale and den Nijs (TKNN) [66] first showed that the quantum number n in the Hall conductance σxy = ne2/h, which is called the TKNN invariant by physicists, can be topologically identified with the first Chern number. In this subsection, we will review the first Chern number of the quantum Hall effect based on topological band theory. For the 2d square lattice Γ ≃ Z2 ⊂ R2, its Pontryagin dual is the 2-torus T2, and it will be called the Brillouin torus from now on. In condensed matter physics, a space dual to the original lattice is called the Brillouin zone B. In physics parlance, this is the momentum space in which the crystal wave vector k lives. 2m2 mω which takes the same form as the quantum harmonic oscillator. Therefore, the 1630003-6 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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