  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 $\eta$.
