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2.3. Dirac Hamiltonian The quantum Hall system can be viewed as a topological insulator, whose effective Hamiltonian is given by the Dirac Hamiltonian. A phase transition is then con- trolled by some parameters in the Dirac Hamiltonian. In order to illustrate such phenomenon, we analyze 2-band models generalizing the monopole (see below) that depend on a function h : B → R3, where B is usually the Brillouin zone and in our case, to be concrete, the Brillouin torus T2. An element k = (kx,ky) ∈ T2 is called a crystal wave vector. A particular choice of h, which we discuss, exhibits a phase transition and will also appear again in the study of the quantum spin Hall effect. Consider the following Hamiltonian depending on k ∈ R3: H=k·σ (2.10) where the σi are the Pauli matrices. This Hamiltonian is non-degenerate outside k = 0. Thus restricting it to S2 ⊂ R2 we are in the situation of the previous subsection, with the Bloch bundle being the sum of two line bundles L1 ⊕ L2. It is a nice calculation to show that the Chern numbers are ±1 in this case [14, 18]. The degenerate point at k = 0 is thought of as a monopole which is responsible for the “charges” given by the first Chern numbers. There is a generalization to a N -band model, where now the σi are spin S = N −1 2 matrices. In this case the Chern numbers are n = 2m with m = −S,−S + 1,...,S [14, 18]. More generally, consider the 2-level Hamiltonian H = hx(k)σx + hy(k)σy + hz(k)σz = h(k) · σ. (2.11) Here σ = (σx,σy,σz), where the σi are the Pauli matrices. Assume that H is nowhere degenerate, i.e. hi(k) ̸= 0, the first Chern number in this case is a winding number. This goes back to [18] and is explained in detail in Appendix A [25]. To see this, define the function: d:T2→S2, d(k)= h(k). |h(k)| The winding number of d computes the first Chern number of the lowest band, aka. the ground state. 1 4π T2 This is the pull back via d of the 2-sphere S2-winding number around the monopole of H = k · σ at k = (0, 0, 0), see [18, 65]. This also works for general Brillouin zones. The first Chern number of the other band is then −c1 since the sum of the two bands is a trivial rank 2 bundle. The particular choice of h(kx,ky) yields: H(k)=Asin(kx)σx +Asin(ky)σy +[M−B(sin2(kx/2)+sin2(ky/2))]σz (2.13) Notes on topological insulators c1 = (∂kxd×∂kyd)·d dk. (2.12) 1630003-9 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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