Bosonization study of a generalized statistics model with four Fermi points

TL;DR

This paper investigates a one-dimensional fractional statistics model with four Fermi points, revealing rich low-energy behaviors including decoupled Luttinger liquids, RG flow to fixed points, and relevance of various order parameters.

Contribution

It introduces a bosonization analysis of a generalized fractional statistics model, uncovering its low-energy structure and potential phase transitions depending on parameters.

Findings

Model exhibits four Fermi points influenced by statistical parameter and filling.

Low-energy modes form two decoupled Tomonaga-Luttinger liquids with variable parameters.

System may flow to a non-trivial fixed point under renormalization group analysis.

Abstract

We study a one-dimensional lattice model of fractional 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 $\phi \ne 0$. 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 non-trivial 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 fractional statistics have a rich structure as a function of $\phi$ and $\eta$.