We present the exact Bethe ansatz solution for the two-dimensional BCS pairing Hamiltonian with p x + ip 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-Read͒ pairs and characterized by a topological order parameter. DOI: 10.1103/PhysRevB.79.180501 PACS number͑s͒: 74.20.Fg, 74.20.Mn, 74.20.Rp In 1957, Bardeen, Cooper, and Schrieffer 1 ͑BCS͒ published an epoch defining paper giving a microscopic explanation of the properties of superconducting metals at low temperatures. The model was based on a reduced Hamiltonian which describes the pairing interaction between conduction electrons. The original study of the BCS model was formulated in the grand-canonical ensemble and solved with a mean-field approximation. In 1963 Richardson 2 derived the exact solution of the reduced BCS Hamiltonian with s-wave symmetry in the canonical ensemble. This solution was largely unnoticed until its rediscovery in the theoretical studies of ultrasmall metallic grains in the 1990s, where it was employed to understand the crossover between the fluctuation dominated regime and the fully developed superconducting regime ͑for a review see Ref. 3͒. The exact solution for the s-wave BCS model is related to the Gaudin spin Hamiltonians, and their integrability can be understood in the general framework of the quantum inverse scattering method.4,5 These later developments allowed for an exact computation of various correlators, 4,6,7 and led to generalizations of the Richardson-Gaudin models with applications to condensed matter and nuclear physics. 3,8 In this Rapid Communication we analyze the twodimensional ͑2D͒ BCS model where the symmetry of the pairing interaction is p x + ip y ͑hereafter referred to as p + ip͒. The Hamiltonian of the model is ͑1͒where c k , c k † are destruction and creation operators of 2D spinless or polarized fermions with momentum k, m is their mass, and G is a dimensionless coupling constant which is positive for an attractive interaction. The p + ip model has attracted considerable attention due to the connection with the Moore-Read ͑MR͒ pfaffian state arising in the quantum Hall effect at filling fraction 5/2, 9 which has been proposed to support non-Abelian anyons allowing for topological quantum computation. 10,11 Motivated by these considerations, concre...
Using the well-known trigonometric six-vertex solution of the Yang-Baxter equation we derive an integrable pairing Hamiltonian with anyonic degrees of freedom. The exact algebraic Bethe ansatz solution is obtained using standard techniques. From this model we obtain several limiting models, including the pairing Hamiltonian with p + ip-wave symmetry. An in-depth study of the p + ip model is then undertaken, including a mean-field analysis, analytic and numerical solution of the Bethe ansatz equations, and an investigation of the topological properties of the ground-state wavefunction. Our main result is that the ground-state phase diagram of the p + ip model consists of three phases. There is the known boundary line with gapless excitations that occurs for vanishing chemical potential, separating the topologically trivial strong pairing phase and the topologically non-trivial weak pairing phase. We argue that a second boundary line exists separating the weak pairing phase from a topologically trivial weak coupling BCS phase, which includes the Fermi sea in the limit of zero coupling. The ground state on this second boundary line is the Moore-Read state.Combining these we have a single determinant expression
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