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.
One of our main tenets is that progress in understanding these lattice models can be achieved if one clearly distinguishes between approximations justified by physical arguments and manipulations that are mathematically exact (rigorous); the present paper belongs to the latter category. We mention that the model derived in @cite_0 corresponds to the special case @math ; the generalization to different coupling parameters, allowing also for negative values, is natural from a mathematical point of view. We also note that, while an outline of how to diagonalize the Mattis Hamiltonian by bosonization was already given in @cite_0 , Section 6.2, the details and complete solution of the model are given here for the first time.
@cite_0
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A detailed derivation of a two dimensional (2D) low energy effective model for spinless fermions on a square lattice with local interactions is given. This derivation utilizes a particular continuum limit that is justified by physical arguments. It is shown that the effective model thus obtained can be treated by exact bosonization methods. It is also discussed how this effective model can be used to obtain physical information about the corresponding lattice fermion system.