Strings in two-dimensional classicalXYmodels

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.