  We construct models of interacting itinerant non-Abelian anyons moving along
one-dimensional chains. We focus on itinerant Ising (Majorana) and Fibonacci
anyons, which are, respectively, related to SU(2)_2 and SU(2)_3 anyons and
also, respectively, describe quasiparticles of the Moore-Read and
Z_3-Read-Rezayi fractional quantum Hall states. Following the derivation of the
electronic large-U effective Hubbard model, we derive effective anyonic t-J
models for the low-energy sectors. Solving these models by exact
diagonalization, we find a fractionalization of the anyons into charge and
(neutral) anyonic degrees of freedom -- a generalization of spin-charge
separation of electrons which occurs in Luttinger liquids. A detailed
description of the excitation spectrum can be performed by combining spectra
for charge and anyonic sectors. The anyonic sector is the one of a squeezed
chain of localized interacting anyons, and hence is described by the same
conformal field theory (CFT), with central charge c=1/2 for Ising anyons and
c=7/10 or c=4/5 for Fibonacci anyons with antiferromagnetic or ferromagnetic
coupling, respectively. The charge sector is the spectrum of a chain of
hardcore bosons subject to phase shifts which coincide with the momenta of the
combined anyonic eigenstates, revealing a subtle coupling between charge and
anyonic excitations at the microscopic level (which we also find to be present
in Luttinger liquids), despite the macroscopic fractionalization. The combined
central charge extracted from the entanglement entropy between segments of the
chain is shown to be 1+c, where c is the central charge of the underlying CFT
of the localized anyon (squeezed) chain.
