We present the exact Bethe ansatz solution for the two-dimensional BCS pairing Hamiltonian with $p_x+i{p}_y$ symmetry. Using both mean-field theory and the exact solution we obtain the ground-state phase diagram parametrized by the filling fraction and the coupling constant. It consists of three phases that are denoted weak-coupling BCS, weak pairing, and strong pairing. The first two phases are separated by a topologically protected line where the exact ground state is given by the Moore-Read pfaffian state. In the thermodynamic limit the ground-state energy is discontinuous on this line. The other two phases are separated by the critical line, also topologically protected, previously found by Read and Green. We establish a duality relation between the weak and strong pairing phases, whereby ground states of the weak phase are “dressed” versions of the ground states of the strong phase by zero energy (Moore-Read) pairs and characterized by a topological order parameter.

• Received 15 April 2009

DOI:https://doi.org/10.1103/PhysRevB.79.180501