Continuum and low energy limit of one-dimensional lattice anyons

We consider interacting abelian anyons on a one-dimensional lattice which can be realized in ultracold gases experiments with density dependent hopping. It is shown that the continuum limit is possible even while maintaining the full topological exchange properties. The theory differs from previous 1D continuous anyon representations since the 2π periodicity of the anyonic phase is maintained, which normally is only possible for 2D anyons with continuous topological exchange. We derive the corresponding low energy Luttinger Liquid theory, which has the hallmark of different velocities for left- and right-moving collective excitations.