N-Dependent Fractional Statistics ofNVortices

Abstract

A system of $N$ identical point vortices on a plane, e.g., a superfluid-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ thin film, is studied with use of elementary quantum mechanics. We find the surprising result that the system obeys fractional or $\theta$ statistics which depends on $N$, where $\theta \mathrm{}\left(N\right)=\frac{\pi }{N}$ or $\frac{\pi }{N}+\pi$. This result arises from the angular momentum contained in the zero-point motion of the vortices.