Jordan-Wigner transformation for quantum-spin systems in two dimensions and fractional statistics

Abstract

I construct a Jordan-Wigner transformation for spin-one-half quantum systems on two-dimensional lattices. I show that the spin-one-half XY (i.e., a hard-core Bose system) is equivalent (on any two-dimensional Bravais lattice) to a system of spinless fermions and gauge fields satisfying the constraint that the gauge flux on a plaquette must be proportional to the spin (particle) density on site. The constraint is enforced by the addition of a Chern-Simons term of strength θ to the Lagrangian of the theory. For the particular value θ=1/2π, the resulting particles are fermions. In general they are anyons. The implications of these results for quantum spin liquids are briefly discussed.