New exactly solvable models of interacting bosons and fermions

Richardson's exact solution of the pairing model can be generalized to three families of exactly solvable models for interacting bosons and fermions. We focus on the rational family and show how to map these models onto classical two-dimensional electrostatic problems. In the case of fermions, we use the electrostatic mapping of the pairing model to provide a new perspective on the superconducting phase transition in finite nuclei. In the case of bosons, we show that this class of models displays a second-order phase transition to a fragmented state in which only the two lowest boson states are macroscopically occupied and suggest that this provides a new mechanism for sd dominance in interacting boson models of nuclei.