  We study pairing of an interacting three component Fermi gas in two
dimensions. By using a mean field theory to decouple the interactions between
different pairs of Fermi components, we study the free energy landscapes as a
function of various system parameters including chemical potentials, binding
energies, and temperature. We find that the s-wave pairing channel is
determined by both chemical potentials and the interaction strengths between
the three available channels. We find a second order thermal phase transition
and a series of first order quantum phase transitions for a homogenous system
as we change the parameters. In particular, for symmetric parameters, we find
the simultaneous existence of three superfluid orders as well as re-entrant
quantum phase transitions as we tune the parameters.
