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In the classical Hall system, a magnetic field B is turned on perpendicular to a plane of finite size. Then one applies an electric current and considers a single elec- tron flowing in one direction in the plane, say the y-direction. The Hall conductance is computed as the ratio of the current density to the electric field, σxy=jy =νe2 Ex h where ν is the so-called filling factor, e is the elementary charge and h is the Planck constant. When the magnetic field is very strong and the temperature is low enough, the filling factor is quantized and takes on positive integer values. This phenomenon is called the integer quantum Hall effect, e2 σxy =nh, n∈N. (2.1) The quantum number n, also called the TKNN integer, is successfully understood as the first Chern number of the Berry curvature over the Brillouin torus. For an even stronger magnetic field and at extremely low temperature, the fractional quantum Hall effect can be observed, in which the filling factor can also take on fractional values. The fractional QHE will not be discussed here, since we will only consider Hamiltonians with weak interactions in this survey while in the fractional case the Coulomb interaction between electrons is dominant. 2.1. Landau levels As a warm up, we consider the spectral theory of a free Hamiltonian in the Landau gauge in this subsection. Its spectrum consists of the Landau levels, which is one of the simplest and most important band structures in condensed matter physics. Let us first forget about the spin property of electrons for the moment. A free electron in a magnetic field has the Hamiltonian H= 1(p+eA)2. 2m Since the magnetic field B is assumed to be perpendicular to the xy-plane, we consider the Landau gauge A = (0,Bx,0) such that B = ∇×A = (0,0,B), in this gauge H = 1 [ p 2x + ( p y + e B x ) 2 ] . 2m It is obvious that the commutator [H,py] = 0, so H and py can be diagonalized simultaneously. In other words, in the Hamiltonian py may be replaced by its eigen- value ky, H = 1 [ p 2x + ( k y + e B x ) 2 ] . 2m Notes on topological insulators 1630003-5 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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