Phase diagram of a model for topological superconducting wires

We calculate the phase diagram of a model for topological superconducting wires with local s-wave pairing, spin-orbit coupling $\vec{\lambda}$ and magnetic field $\vec{B}$ with arbitrary orientations. This model is a generalized lattice version of the one proposed by Lutchyn $\textit{et al.}$ [Phys. Rev. Lett. $\textbf{105}$ 077001 (2010)] and Oreg $\textit{et al.}$ [Phys. Rev. Lett. $\textbf{105}$ 177002 (2010)], who considered $\vec{\lambda}$ perpendicular to $\vec{B}$. The model has a topological gapped phase with Majorana zero modes localized at the ends of the wires. We determine analytically the boundary of this phase. When the directions of the spin-orbit coupling and magnetic field are not perpendicular, in addition to the topological phase and the gapped non topological phase, a gapless superconducting phase appears.