A system of N identical point vortices on a plane, e.g., a superfluid-4 He thin film, is studied with use of elementary quantum mechanics. We find the surprising result that the system obeys fractional or 9 statistics which depends on N 9 where 0(N) = TT/N or TT/N + rr. This result arises from the angular momentum contained in the zero-point motion of the vortices.PACS numbers: 03.40. Gc , 67.40.Vs Fractional topological quantum numbers have recently been emerging as a common feature of condensed-matter physics in lower dimensions, and lower-dimensional relativistic field theories. Specifically, solitons with fractional charge and fermion number, 1 and with fractional spin and statistics, 2 have appeared in a number of contexts. A notable example is the \e quasiparticle in the quantum Hall fluid. 3 In a previous paper, 4 we argued that the vortex in a thin film of superfluid, which is a two-dimensional soliton with a topologically nontrivial hole at its center, possesses fractional, or 0, statistics. This we did by studying the quantum dynamics of two identical vortices. We concluded that a single vortex in this system obeys quarter-fractional statistics (i.e., 0=7r/2 or 37r/2). Hence, it is neither a boson (0 = 0) nor a fermion (0 = IT ).Here we generalize this previous result to the case of N identical point vortices on a plane. These N vortices carry N holes, with global consequences. The surprising outcome of this present study is that 9 for this system depends on the number of vortices: We find that 0( N) = TT/AT or TT/N + TT. It has been recently pointed out that the braid group allows representations where 9 depends on the total number of particles. Ringwood and Woodward, 5 in the specific case of 't HooftPolyakov monopoles, have suggested that the set of allowed values of 9 depends on the number of monopoles. More generally, Thouless and Wu 6 have argued that the braid group acting on particles on a sphere leads to a number dependence of the set of possible 9 values. However, in previous models, e.g., anyons, 9 was assumed to be independent of the number of particles. 7 This then, is the first specific dynamical system for which N -dependent statistics is suggested as being necessary.A real physical system of vortices, e.g., in superfluid-4 He thin films, arises nontrivially from the underlying many-body problem. Here, for want of a satisfactory many-body theory in general, we start not from the microscopic picture, but from a macroscopic effective Hamiltonian for point vortices. A justification for this lies in the fact that the KosterlitzThouless theory, which starts with the same Hamiltonian, has been experimentally verified. 8 The key assumptions which we share in common with the
