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.
We choose the name of the model studied in this paper to acknowledge a pioneering paper published by Mattis in 1987 in which he pointed out that a 2D interacting fermion Hamiltonian similar to ours can be mapped exactly to a non-interacting boson Hamiltonian @cite_1 ; the dispersion relations of this boson Hamiltonian were given by Mattis (see @cite_1 , Equation (8)) but no details were provided. Mattis also argued that this model can arise from a tight-binding description of 2D lattice fermions at half-filling with a square Fermi surface and in the absence of nesting-type instabilities, but, as shown in @cite_0 , this latter proposal is doubtful (at half-filling, there are additional interaction terms that cannot be bosonized in a simple manner and which are likely to yield a gap, very different to what the Mattis model describes).
@cite_0 @cite_1
2000443329 2093079154
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. We consider the tight-binding energy band in sq lattices and determine that in the half-filled case there exists an infrared instability in addition to the 2k sub F (nesting)-type instability. In view of the pseudo-two-dimensional band structure of La sub 2- sub x Ba sub x CuO sub 4 recently proposed by and by Mattheiss, our conclusions should be relevant to this material at least, one of which demonstrates the remarkable phenomenon of high-T sub c superconductivity. To further analyze it, we devise a two-dimensional bosonization scheme. Three-dimensional bosonization is also briefly discussed.