  We compute mean field phase diagrams of two closely related interacting
fermion models in two spatial dimensions (2D). The first is the so-called 2D
t-t'-V model describing spinless fermions on a square lattice with local
hopping and density-density interactions. The second is the so-called 2D
Luttinger model that provides an effective description of the 2D t-t'-V model
and in which parts of the fermion degrees of freedom are treated exactly by
bosonization. In mean field theory, both models have a charge-density-wave
(CDW) instability making them gapped at half-filling. The 2D t-t'-V model has a
significant parameter regime away from half-filling where neither the CDW nor
the normal state are thermodynamically stable. We show that the 2D Luttinger
model allows to obtain more detailed information about this mixed region. In
particular, we find in the 2D Luttinger model a partially gapped phase that, as
we argue, can be described by an exactly solvable model.
