We study a class of simple, translationally invariant, two-dimensional, nearest-neighbor, isotropic XY models which possess, in addition to the familiar integer vortices, half-integer vortex and ‘‘string’’ excitations. The half-integer vortices interact through both the logarithmic Coulomb potential and a linear potential mediated by the strings. The phase diagram of these models consists of three phases separated by lines of conventional Kosterlitz-Thouless transitions and lines of Ising transitions driven by the vanishing of the tension in the strings.

• Received 14 March 1985

DOI:https://doi.org/10.1103/PhysRevLett.55.541