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We study Read and Green's mean-field model of the spinless $p_x+ip_y$superconductor [N.Read and D.Green, Phys. Rev. B 61, 10267 (2000)] at a specialset of parameters where we find the analytic expressions for the topologicallydegenerate ground states and the Majorana modes, including in finite systemswith edges and in the presence of an arbitrary number of vortices. Thewavefunctions of these ground states are similar (but not always identical) tothe Moore-Read Pfaffian states proposed for the $\nu=5/2$ fractional quantumHall system, which are interpreted as the p-wave superconducting states ofcomposite fermions. The similarity in the long-wavelength universal propertiesis expected from previous work, but at the special point studied herein thewavefunctions are exact even for short-range, non-universal properties. As anapplication of these results, we show how to obtain the non-Abelian statisticsof the vortex Majorana modes by explicitly calculating the analyticcontinuation of the ground state wavefunctions when vortices are adiabaticallyexchanged, an approach different from the previous one based on universalarguments. Our results are also useful for constructing particlenumber-conserving (and interacting) Hamiltonians with exact projectedmean-field states.

Analytic ground state wavefunctions of mean-field $p_x + ip_y$ superconductors with vortices and boundaries