  In this work we theoretically study pairing in two-dimensional Fermi gases, a
system which is experimentally accessible using cold atoms. We start by
deriving the mean-field pairing gap equation for a coordinate-space potential
with a finite interaction range, and proceed to solve this numerically. We find
that for sufficiently short effective ranges the answer is identical to the
zero-range one. We then use Diffusion Monte Carlo to evaluate the total energy
for many distinct particle numbers; we employ several variational parameters to
produce a good ground-state energy and then use these results to extract the
pairing gap across a number of interaction strengths in the strongly
interacting two-dimensional crossover. Extracting the gap via the odd-even
energy staggering, our microscopic results can be used as benchmarks for other
theoretical approaches.
