The Average Field Approximation for Almost Bosonix Extended Anyons

Abstract

Anyons are 2D or 1D quantum particles with intermediate statistics, interpolating between bosons and fermions. We study the ground state of a large number N of 2D anyons, in a scaling limit where the statistics parameter is proportional to the inverse of N. This means that the statistics is seen as a "perturbation from the bosonic end." We model this situation in the magnetic gauge picture by bosons interacting through long-range magnetic potentials. We assume that these effective statistical gauge potentials are generated by magnetic charges carried by each particle, smeared over discs of radius R (extended anyons). Our method allows to take R to 0 not too fast at the same time as N to infinity. In this limit we rigorously justify the so-called "average field approximation": the particles behave like independent, identically distributed bosons interacting via a self-consistent magnetic field.