Life at the edge: Fermionic superfluidity in ultracold atomic gases near a boundary

Theoretically, for ultracold atomic gases in the many-body regime it is common to approximate the microscopic van der Waals potential by a delta-function with an unspecified coupling constant. This unknown coupling constant can usually be related to a measurable physical parameter; in ultracold gases this is done by connecting it to the s-wave scattering length. However, an additional technicality occurs with a delta-function interaction in dimensions greater than one. In two- and three-dimensions, a short-ranged attractive interaction leads to unphysical divergences in various quantities, for example, the bound state energy or the superfluid pairing amplitude become divergent. It is well-known that in the bulk, far away from any boundaries, these divergences can be systematically healed via a renormalization of the coupling constant, but for trapped systems or near a boundary there is no clear-cut prescription to heal these divergences and simultaneously satisfy arbitrary boundary conditions. In this talk, I will discuss some prior work on solving this problem for the local superfluid pairing, as well as some preliminary results of our own. For a fermionic superfluid, we formalize how to regularize the local pairing amplitude and satisfy a given boundary condition. Applying this formulation to a fermionic superfluid with single hard-wall boundary, we find a large enhancement of the superfluid pairing near the boundary, but essentially no change in the superfluid transition temperature, which is something that has been predicted to occur in a superconducting system.