Zero-energy bound state trapped in line-shaped vortex in topological superconductor

Fermion bound states in the core of a line-shaped vortex of a two-dimensional topological superconductor are investigated. The superconducting pairing potential, described in terms of elliptical coordinates, vanishes along a line defect with the two foci at the endpoints. The superconductivity is induced into a topological insulator via proximity effect with a type II s-wave superconductor. The spin and the momentum are perpendicularly locked by the strong spin-orbit coupling via Rashba interaction. A zero-energy Majorana state arises from the Berry phase together with a sequence of equally spaced fermion exitations. By solving the Bogoliubov-de Gennes equations using the method employed by Caroli, de Gennes and Matricon we calculate the energies, the wave-functions and spin-polarization of the bound states. An analytic expression for the local density of states within the vortex is obtained.