Calculation of the topological numbers of two dimensional topological superconductors

The topological structures of a two dimensional topological superconductor with a spin-orbital coupling is studied. Trivial and non-trivial topological regions are separated by the curves where the band gap closes. The Majorana topological number M(H) proposed by Kitaev for one dimensional quantum wires is expanded to the two dimensional case. The Majorana topological number of the above superconductor is calculated and three topological phase diagrams are determined. These diagrams are fully in accordance with those given by TKNN parity. The method of the former is much simpler. A two dimensional and infinite slice with a width of 24 lattice sites is calculated to show the Majorana zero modes. Using a set of parameters inside the non-trivial topological regions of the phase diagrams a zero energy appears at k_x=\pi and has a wave function near the edges of the slice(Majorana zero mode). For parameters outside the non-trivial topological regions, however, the band has either no zero mode or two zero modes, which are not topologically stable.