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.
The present paper is the third in a series @cite_0 @cite_2 in which we propose and develop a method to do reliable, non-perturbative computations in models of strongly interacting fermions on a 2D square lattice @cite_49 . We have so far considered the simplest non-trivial case, the so-called 2D model, describing spinless fermions with local interactions. In @cite_0 , a 2D analogue of the Luttinger model was derived from the 2D model using a particular partial continuum limit. The interested reader can find more details in Appendix . This 2D Luttinger model is an extension of the Mattis model by so-called that cannot be bosonized. In @cite_2 we showed that, within a mean field approximation, there is a non-trivial phase away from half-filling in which the antinodal fermions are gapped, and in this phase the 2D Luttinger model reduces to the Mattis model. One key ingredient of our approach is the notion of underlying Fermi surface arcs (not necessarily corresponding to a true Fermi surface) in the 2D model away from half filling. Previous theoretical work predicting four disconnected Fermi surface arcs in 2D interacting lattice fermion systems include renormalization group studies in @cite_39 @cite_41 .
@cite_41 @cite_39 @cite_0 @cite_49 @cite_2
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We study a two-dimensional Fermi liquid with a Fermi surface containing the saddle points @math and @math . Including Cooper and Peierls channel contributions leads to a one-loop renormalization group flow to strong coupling for short range repulsive interactions. In a certain parameter range the characteristics of the fixed point, opening of a spin and charge gap and dominant pairing correlations are similar to those of a 2-leg ladder at half-filling. An increase of the electron density we argue leads to a truncation of the Fermi surface with only 4 disconnected arcs remaining. 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 present a fermion model that is, as we suggest, a natural 2D analogue of the Luttinger model. We derive this model as a partial continuum limit of a 2D spinless lattice fermion system with local interactions and away from half filling. In this derivation, we use certain approximations that we motivate by physical arguments. We also present mathematical results that allow an exact treatment of parts of the degrees of freedom of this model by bosonization, and we propose to treat the remaining degrees of freedom by mean field theory. We compute mean field phase diagrams of two closely related interacting fermion models in two spatial dimensions (2D). The first is the so-called 2D t-t′-V model describing spinless fermions on a square lattice with local hopping and density-density interactions. The second is the so-called 2D Luttinger model that provides an effective description of the 2D t-t′-V model and in which parts of the fermion degrees of freedom are treated exactly by bosonization. In mean field theory, both models have a charge-density-wave (CDW) instability making them gapped at half-filling. The 2D t-t′-V model has a significant parameter regime away from half-filling where neither the CDW nor the normal state are thermodynamically stable. We show that the 2D Luttinger model allows to obtain more detailed information about this mixed region. In particular, we find in the 2D Luttinger model a partially gapped phase that, as we argue, can be described by an exactly solvable model.