Exactly solvable Richardson-Gaudin models in condensed matter and nuclear physics

Abstract

The exact solution of the SU(2) pairing Hamiltonian with non-degenerate single particle orbits was introduced by Richardson in the early sixties. It passed largely unnoticed till it was rediscovered in recent years and widely applied to mesoscopic systems. In this talk I will review the wide class of exactly solvable pairing Hamiltonians that can be derived from the SU(2) Richardson-Gaudin (RG) integrable models. The rational family of RG models leads to s-wave pairing Hamiltonians whose exact wavefunction unveils the unique structure the Cooper pairs and shows how they evolve along the crossover from BCS to BEC. On the contrary, the hyperbolic family of RG models realizes p-wave pairing Hamiltonians with topological phases and quantum phase transitions, as well as more realistic separable pairing Hamiltonians for atomic nuclei. Then, I will show how the Richardson-Gaudin models could be extended to larger rank algebras like SO(5) and SO(8) to describe proton-neutron pairing Hamiltonians or SO(6) for color pairing. These Hamiltonians not only constitute excellent benchmark models to test many-body approximations, but they can also suggest ways to treat many-body correlations.