Exactly solvable pairing models in two dimensions

The BCS theory models electron correlations with pure zero-momentum pairs. Here we consider a family of pairing Hamiltonians, where electron correlations are modelled by pure arbitrary-momentum pairs. These models are two-dimensional, defined by S_z which is the only conserved component of the total electron spin. We find that all models in this family are exactly solvable, and present the solutions. It is interesting to note that the d -wave pair condensate state in high T_c superconductivity could be a ground state of one of model Hamiltonians, which has analytical solutions. Significantly, it is shown that there is a sudden change in ground state energy versus pairing strength, suggesting a quantum phase transition from an independent-particle picture to the d -wave pair condensate. (paper)