Dual view of the 3D Heisenberg model and the Abelian projection

Abstract

The Heisenberg model in 3D is studied from a dual point of view. It is shown that it can have vortex configurations, carrying a conserved charge [U(1) symmetry]. Vortices condense in the disordered (demagnetized) phase. A disorder parameter $〈\ensuremath{\mu}〉$ is defined, dual to the magnetization $〈\stackrel{\ensuremath{\rightarrow}}{n}〉,$ which signals the condensation of vortices, i.e., spontaneous breaking of dual U(1) symmetry. This study sheds light on the procedure known as Abelian projection in non-Abelian gauge theories.