Fractional vortices, Z2 gauge theory, and the confinement-deconfinement transition

Abstract

In this Letter we discuss the classical three-dimensional XY model whose nearest-neighbor interaction is a mixture of $cos({\ensuremath{\theta}}_{i}\ensuremath{-}{\ensuremath{\theta}}_{j})$ (ferromagnetic) and $cos2({\ensuremath{\theta}}_{i}\ensuremath{-}{\ensuremath{\theta}}_{j})$ (nematic). This model is dual to a theory with integer and half-integer vortices. While both types of vortices interact with a noncompact $U(1)$ gauge field, the half-integer vortices interact with an extra interaction mediated by a ${\mathbb{Z}}_{2}$ gauge field. We shall discuss the confinement-deconfinement transition of the half-integer vortices, the Wilson and the `t Hooft loops, and their mutual statistics in path integral language. In addition, we shall present a quantum version of the classical model which exhibits these physics.