Topological frustration: Quantum statistics of vortices from a dual theory of the XY ferromagnet

The familiar concept of geometrical frustration can be extended to frustration induced by the topology of spin textures. For instance, in a magnetic vortex the spins on the periphery rotate in the easy plane, leaving the spins at the vortex core frustrated. To reduce the exchange energy, the core spins come out of the easy plane and point along the hard axis. There is a degeneracy resulting from the associated Z₂ symmetry (polarization) which has well-known important consequences for the dynamics of a vortex. Here we show that topological frustration at the vortex core may induce a nontrivial quantum statistics of vortices. We extend the well-known mapping between the easy-plane ferromagnet and electrostatics in d = 2 spatial dimensions, introduced by Kosterlitz [1], to dynamical and quantum phenomena in a d = 2+1 spacetime. Ferromagnetic vortices behave like quantum particles with an electric charge equal to the vortex number and a magnetic flux equal to the transverse spin of the vortex core. Vortices with half-integer core spin exhibit fermionic statistics.

[1] J. M. Kosterlitz, J. Phys. C 7, 1046 (1974).