Exactly soluble model of fractional statistics.

Abstract

An anyon is a particle in 2+1 dimensions obeying fractional statistics' and may be viewed as a hard-core boson (or a fermion) to which a flux tube is attached. The statistical interaction is highly nontrivial and it has so far resisted full solution. Some progress can be made using a mean-field theory of statistics in which the sta- tistical flux is replaced by a uniform magnetic field 8,&. This idea was first put to practical use in the theory of the fractional quantum Hall effect (FQHE), where fermions in an external field 8,, can be mapped onto bo- sons in zero mean field (B,tt+8, =0). The ordering associated with the FQHE gap is revealed in the form of algebraic of-diagonal long-range order (ODLRO) for the composite particles (bosons plus flux tubes). More recently this idea has been greatly expanded and extend- ed by Laughlin ' to the case of semions (particles with statistical angle 8/tr = 1/2) for which the mean-field theory yields fermions filling the two lowest Landau lev- els. Laughlin pointed out the remarkable fact that pairs of semions can condense to form a Bose superfluid. The mean-field approximation has several peculiari- ties ' since the purely quantum Aharonov-Bohm phase is replaced by a classical Lorentz force and the particles see a preferred (quantum) length scale I— = (Ac/eB, ff) ' This turns out not to be a problem in the FQHE since the incompressible Laughlin state has a gap and its den- sity is pinned at precisely this same length scale due to the (physical) external field. However, mean-field theory for semions incorrectly suggests the existence of an excitation gap analogous to that in the integer Hall effect. Augmenting the mean-field theory with fluctua- tions at the RPA level'' restores the linearly dispersing density mode expected in a superfluid. A local increase in density is accompanied by a compensating increase in the local B,g such that the lowest two Landau levels can remain filled and there is no preferred length scale. ' The purpose of this paper is to present an exactly solu- ble model of anyons which sheds considerable light on all these points and which totally bypasses difficulties in- herent in the mean-field solutions. The model differs in one (highly) nontrivial way from the usual one in that the particles have an attractive hard-core interaction. (The effect of the hard-core repulsion that is usually con- sidered is to prevent intersection of the boson world lines and hence make the homotopy class of their braiding well defined. ') The present model was inspired by recent applications of supersymmetry and the Atiyah-Singer in- dex theorem to particles in an arbitrary magnetic field. We will prove below that the index theorem can be ex- tended to the nontrivial many-body case where the flux is not axed in time but is carried on the particles them selves; i.e., for anyons. Consider the pair of Hamiltoni- ans .'V