Luttinger liquids, Fermi liquids, and fractional statistics

We discuss how one-dimensional interacting fermion systems, which in the low-energy approximation are described by Luttinger liquid theory, can be reformulated as systems of weakly interacting particles with fractional charge and statistics. Our approach is to use Landau's phenomenological approach to Fermi liquid theory, where the quasiparticles are interpreted as adiabatically dressed fermions. In an earlier publication, the local charge carried by these excitations has been shown to be fractional. We focus here on the statistics of the quasiparticles and show that by a change of momentum variables the Landau parameters of the generalized Fermi fluid can be transformed to zero. This change in interaction is compensated by a change of the entropy function, which is consistent with the interpretation of the quasiparticles as satisfying generalized exclusion statistics.

Figure 1

The single-particle energy ${\tilde{\epsilon }}_0$, shown as function of the momentum variable $k$, with quadratic dispersion and with statistics parameter $\lambda =0.5$. Filled circles correspond to occupied states and open circles to unoccupied states. Two cases are shown. The first case (red circles, long dashed curve) corresponds to a filled Fermi sea, representing the ground state with $N_0=23$ particles. In the second case (blue circles, short dashed curve), two particles are added in the lowest available momentum states close to the Fermi point $k_F$. This corresponds to the case $N=J=2$. The generalized Pauli exclusion between the particles gives rise to a relative shift of the two curves along the $k$ axis.