February 7, 2012 16:30

2D two-bands insulators breaking time reversal symmetry can present topological phase characterized by a topological invariant called the Chern number. In absence of disorder, for periodic system, this Chern number Ch is usually computed as a full Brillouin zone surface integral of the so called Berry curvature. In this talk, we present another way to compute this Chern number, as the discrete sum over a set the Dirac points (this corresponds to the Brouwer degree of the corresponding Hamiltonian map). This perspective allows us to develop a k-space geometrical procedure to engineer Chern insulator models that can be tuned through five topological distinct phases Ch=+/-2,+/-1,0. We present two examples. For each of these models, we discuss the bulk-edge correspondence and we also explain how to construct the corresponding Z2 time-reversal topological insulator.