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Phases in two dimensional $p_x+ip_y$ superconducting systems with next-nearest-neighbor interactions

A chiral $p_x+ip_y$ superconductor on a square lattice with nearest and next-nearest hopping and pairing terms is considered. Gap closures, as various parameters of the system are varied, are found analytically and used to identify the topological phases. The phases are characterized by Chern numbers (ranging from -3 to 3), and (numerically) by response to introduction of weak disorder, edges, and magnetic fields in an extreme type-II limit, focusing on the low-energy modes (which presumably become zero-energy Majorana modes for large lattices and separations). Several phases are found, including a phase with Chern number 3 that cannot be thought of in terms of a single range of interaction, and phase with Chern number 2 that may host an additional, disorder resistant, Majorana mode. The energies of the vortex quasiparticle modes were found to oscillate as vortex position varied. The spatial length scale of these oscillations was found for various points in the Chern number 3 phase which increased as criticality was approached.