First- and Second-Order Phase Transitions in thed=2XYModel

The $d=2\mathrm{}$ $\mathrm{XY}$ model is generalized to include an operator that couples directly to the vortex core energy, while maintaining the full symmetry of the original model. In addition to the usual Kosterlitz-Thouless transition, the model exhibits a true first-order transition and a pseudo first-order transition. The excitations of the model include fractionally charged vortices. Implications for the $d=2\mathrm{}$ melting problem are discussed.