Unitary fermions and Luscher's formula on a crystal

Understand

We consider the low-energy particle-particle scattering properties in a periodic simple cubic crystal.

• In particular, we investigate the relation between the two-body scattering length and the energy shift experienced by the lowest-lying unbound state when this is placed in a periodic finite box.
• We introduce a continuum model for s-wave contact interactions that respects the symmetry of the Brillouin zone in its regularisation and renormalisation procedures, and corresponds to the na{\"i}ve continuum limit of the Hubbard model.
• The energy shifts are found to be identical to those obtained in the usual spherically symmetric renormalisation scheme upon resolving an important subtlety regarding the cutoff procedure.