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.
Mattis' insight described above has received little attention up to now, and it was re-discovered in 1994 independently by Hlubina @cite_33 and Luther @cite_37 . Hlubina presented a model similar to Mattis' that he claimed can be mapped to a non-interacting boson Hamiltonian, and he diagonalized the latter. He also gave phenomenological arguments suggesting that this model describes a two-dimensional Luttinger liquid. However, from his discussion, it is not clear what the solvable model actually is (the model formulated in @cite_33 , Equation (1), is not exactly solvable but needs to be modified --- the necessary changes are similar to ones discussed in detail in @cite_0 ). Luther studied spinfull fermions with a square Fermi surface and argued that the non-interacting part of the Hamiltonian can be simplified to one that can be bosonized exactly, and which is essentially the same as in Mattis' model. Luther diagonalized the bosonized Hamiltonian that one obtains by also including density-density interactions, initially restricting to interactions between opposite Fermi surface faces only @cite_37 (this restriction corresponds to @math in our notation). This was later extended to also include interactions between adjacent faces in @cite_9 .
@cite_0 @cite_9 @cite_37 @cite_33
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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. Interacting electrons with a square Fermi surface are investigated from a bosonic point of view taking into account electron scattering between all faces of the square. Fermion operators are classified according to their dimensions and the stability of the boson fixed-point is investigated. In particular we find, in contrast to previous studies, that the square Fermi surface is unstable to doping in the case of no spin gap and microscopic Hubbard interactions. Electronic states near a square Fermi surface are mapped onto quantum chains. Using boson-fermion duality on the chains, the bosonic part of the interaction is isolated and diagonalized. These interactions destroy Fermi-liquid behavior. Nonboson interactions are also generated by this mapping, and give rise to an alternate perturbation theory about the boson problem. A case with strong repulsions between parallel faces is studied and solved. This solution discards irrelevant operators in a new interaction Hamiltonian. There is spin-charge separation and the square Fermi surface remains square under doping. At half-filling, there is a charge gap and insulating behavior together with gapless spin excitations. This mapping appears to be a general tool for understanding the properties of interacting electrons on a square Fermi surface. We consider spinless electrons in two dimensions with the bare spectrum @math . In momentum space, the interactions among electrons have a finite range @math , which is small compared to the Fermi momentum. A golden rule calculation of the electron lifetime indicates a breakdown of the Landau Fermi liquid in the model. At the one-loop level of perturbation theory, we show that the density wave and the superconducting instabilities cancel each other and there is no symmetry breaking. We solve the model via bosonization; the excitation spectrum is found to consist of gapless bosonic modes as in a one-dimensional Luttinger liquid.