  Using a formalism based on the two-body S-matrix we study two-dimensional
Bose and Fermi gases with both attractive and repulsive interactions.
Approximate analytic expressions, valid at weak coupling and beyond, are
developed and applied to the Berezinskii-Kosterlitz-Thouless (BKT) transition.
We successfully recover the correct logarithmic functional form of the critical
chemical potential and density for the Bose gas. For fermions, the BKT critical
temperature is calculated in BCS and BEC regimes through consideration of Tan's
contact.
