Average-field approximation for dilute almost-bosonic anyons

We study the ground state of a system of N two-dimensional trapped almost-bosonic anyons. This setup can equivalently be viewed as bosons interacting through long-range magnetic potentials generated by magnetic charges carried by each particle. These magnetic charges are assumed to be smeared over discs of radius R—a model known as extended anyons. To recover the point-like anyons perspective, we consider the joint limit R -> 0 as N →∞. We rigorously justify the average-field approximation for any radius R that decays polynomially, and even for certain exponentially decaying R. The average-field approximation asserts that the particles behave like independent, identical bosons interacting through a self-consistent magnetic field. Our result significantly improves upon the best-known estimates from [1, 2], and it in particular covers radii that are much smaller than the mean interparticle distance. The proof strategy builds on a recent work on two-dimensional attractive Bose gases by Junge and the author [3].