We study an exactly solvable quantum field theory (QFT) model describing
interacting fermions in 2+1 dimensions. This model is motivated by physical
arguments suggesting that it provides an effective description of spinless
fermions on a square lattice with local hopping and density-density
interactions if, close to half filling, the system develops a partial energy
gap. The necessary regularization of the QFT model is based on this proposed
relation to lattice fermions. We use bosonization methods to diagonalize the
Hamiltonian and to compute all correlation functions. We also discuss how,
after appropriate multiplicative renormalizations, all short- and long distance
cutoffs can be removed. In particular, we prove that the renormalized two-point
functions have algebraic decay with non-trivial exponents depending on the
interaction strengths, which is a hallmark of Luttinger-liquid behavior.Comment: 59 pages, 3 figures, v2: further references added; additional
subsections elaborating mathematical details; additional appendix with
details on the relation to lattice fermion