Parity violation and the mean field approximation for the anyon gas

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. 1 e-mail: [email protected] 2 e-mail: [email protected] It is common knowledge that the statistics of particles in 2+1 dimensions can be altered by adding an appropriate “fictitious” electric charge and magnetic flux to each particle. This gives the particles a so-called “statistical interaction” which is, in fact, not a real interaction; it involves no force. The only effect is in the quantum mechanical phase between processes which differ only in the windings of the particles around each other [1]. Since the statistics of particles is essentially embodied in such phases (particle exchanges have phase ±1 according to whether the particles are bosons or fermions), this statistical interaction is aptly named. Attaching charge q and flux Φ to the particles results in an additional contribution per particle exchange of exp iqΦ/2. Thus, if qΦ is an even or odd multiple of 2π the statistical interaction introduces no phase or a phase −1 per interchange. In the former case it is trivial; in the latter case (corresponding to one unit of flux) the statistics of the underlying particles are changed from Fermi to Bose and vice versa. However for arbirtrary values of qΦ the resulting statistics is neither Fermi nor Bose and we have anyons [2]. There has been much interest recently in the many-anyon system, the anyon gas. The analysis is complicated, because even if the anyons have no conventional interaction, the statistical interaction must nonetheless be treated as such; in fact, one finds that three-body interactions arise. Most discussions of the anyon gas describe the particles as fermions with statistical interaction, so-called fermion-based anyons [3,4,5]. The approach is a meanfield or average-field approximation (MFA), wherein one replaces to lowest order the statistical magnetic field by a uniform one with the same average value. Then the difference between the actual and mean magnetic fields is treated as a perturbation, for instance in the RPA approximation.