A smooth polaron-molecule crossover in a Fermi system

The problem of a single down-spin particle interacting with a Fermi sea of up-spin particles is of current interest in the field of cold atoms. The Hubbard model, appropriate to atoms in an optical lattice potential, is considered in parallel with a gas model. As the strength of an attractive short-range interaction is increased there is a crossover from 'polaron' behaviour, in which the Fermi sea is weakly perturbed, to 'molecule' behaviour in which the down-spin particle is bound to a single up-spin particle. It is shown that this is a smooth crossover, not a sharp transition as claimed by many authors.