We study the ground state of a large number N of 2D anyons in an external magnetic field. We consider a scaling limit where the statistics parameter α tends to zero when N tends to infinity which allows the statistics to be seen as a "perturbation from the bosonic end". Our model is that of bosons in a magnetic field and interacting through long-range magnetic potential generated by magnetic charges carried by each particle, smeared over discs of radius R. Our method allows to take R tends to not too fast at the same time as N tends to infinity. We use the information theoretic version of the de Finetti theorem of Brandão and Harrow to justify the so-called "average field approximation": the particles behave like independent, identically distributed bosons interacting via a self-consistent magnetic field.
Abstract 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 R → 0 as
N→∞ 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].
We examine an approach to justifying the mean field approximation for the anyon gas, using the scattering of anyons. Parity violation permits a nonzero average scattering angle, from which one can extract a mean radius of curvature for anyons. If this is larger than the interparticle separation, one expects that the graininess of the statistical magnetic field is unimportant, and that the mean field approximation is good. We argue that a non-conventional interaction between anyons is crucial, in which case the criterion for validity of the approximation is identical to the one deduced using a self-consistency argument.
In this work we consider the
N-body Hamiltonian describing the microscopic structure of a quantum gas of almost-bosonic anyons. This description includes both extended magnetic flux and spin-orbit/soft-disk interaction between the particles which are confined in a scalar trapping potential. We study a physically well-motivated ansatz for a sequence of trial states, consisting of Jastrow repulsive short-range correlations and a condensate, with sufficient variational freedom to approximate the ground state (and possibly also low-energy excited states) of the gas. In the limit
N→∞, while taking the relative size of the anyons to zero and the total magnetic flux
2πβ to remain finite, we rigorously derive the stationary Chern-Simons-Schrödinger/average-field-Pauli effective energy density functional for the condensate wave function. This includes a scalar self-interaction parameter
γ which depends both on
β, the diluteness of the gas, and the spin-orbit coupling strength
g, but becomes independent of these microscopic details for a particular value of the coupling
g=2 in which supersymmetry is exhibited (on all scales, both microscopic and mesoscopic) with
γ=2π∣β∣. Our findings confirm and clarify the predictions we have found in the physics literature.