Bosonization study of a generalized statistics model with four Fermi points

We study a one-dimensional lattice model of generalized statistics in which particles have next-nearest-neighbor hopping between sites which depends on the occupation number at the intermediate site and a statistical parameter phi. The model breaks parity and time-reversal symmetries and has four-fermion interactions if phi = 0. We first analyze the model using mean field theory and find that there are four Fermi points whose locations depend on phi and the filling eta. We then study the modes near the Fermi points using the technique of bosonization. Based on the quadratic terms in the bosonized Hamiltonian, we find that the low-energy modes form two decoupled Tomonaga-Luttinger liquids with different values of the Luttinger parameters which depend on phi and eta; further, the right- and left-moving modes of each system have different velocities. A study of the scaling dimensions of the cosine terms in the Hamiltonian indicates that the terms appearing in one of the Tomonaga-Luttinger liquids will flow under the renormalization group and the system may reach a nontrivial fixed point in the long distance limit. We examine the scaling dimensions of various charge density and superconducting order parameters to find which of them is the most relevant for different values of phi and eta. Finally, we look at two-particle bound states that appear in this system and discuss their possible relevance to the properties of the system in the thermodynamic limit. Our work shows that the low-energy properties of this model of generalized statistics have a rich structure as a function of phi and eta.