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.
An important question is whether the Mattis model describes a 2D Luttinger liquid @cite_5 . Different regularizations and treatments of Mattis-like models can give different results, as exemplified by Refs. @cite_3 and @cite_44 : these works are on such models that differ in regularization details of the interaction, they use different methods, and they obtain different results. This suggests to us that it is important to be mathematically precise both in the definition of the model and of Luttinger-liquid behavior, as well as in the methods used to study them.
@cite_44 @cite_5 @cite_3
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We study instabilities occurring in the electron system whose Fermi surface has flat regions on its opposite sides. Such a Fermi surface resembles Fermi surfaces of some high- @math superconductors. In the framework of the parquet approximation, we classify possible instabilities and derive renormalization-group equations that determine the evolution of corresponding susceptibilities with decreasing temperature. Numerical solutions of the parquet equations are found to be in qualitative agreement with a ladder approximation. For the repulsive Hubbard interaction, the antiferromagnetic (spin-density-wave) instability dominates, but when the Fermi surface is not perfectly flat, the d-wave superconducting instability takes over. Analysis of interacting fermion systems shows that there are two fundamentally different fixed points, Fermi-liquid theory and «Luttinger-liquid theory» (Haldane), a state in which charge and spin acquire distinct spectra and correlations have unusual exponents. The Luttinger liquids include most interacting one-dimensional systems, and some higher- (especially two) dimensional systems in which the band spectrum is bounded above: systems with Mott-Hubbard gaps and an upper Hubbard band. We give a theory which is useful in calculating normal-state, and some superconducting, properties of high-T c superconductors We consider a system of two-dimensional interacting fermions with a flat Fermi surface. The apparent conflict between Luttinger and non-Luttinger liquid behaviors found through different approximations is resolved by showing the existence of a line of nontrivial fixed points, for the renormalization group (RG) flow, corresponding to Luttinger liquid behavior; the presence of marginally relevant operators can cause flow away from the fixed point. The analysis is nonperturbative and based on the implementation, at each RG iteration, of Ward identities obtained from local phase transformations depending on the Fermi surface side, implying the partial vanishing of the beta function.