Vortices and strings: Phase transition in anisotropic planar-rotor systems

We have investigated the phase transition of a two-dimensional system described by the Hamiltonian H=-J ${\mathcal{J}}_{\mathrm{NN}}$ cos(${\mathrm{theta}}_i$-${\mathrm{theta}}_j$)-K JNN cos(${\mathrm{theta}}_i$+${\mathrm{theta}}_j$+${\phi }_{\mathrm{ij}}$), for J,K>0 and 0≤K/J≤1, and where NN denotes nearest neighbors, which has both Ising-like domain-wall (string) and XY-like vortex excitations. Migdal-Kadanoff and Monte Carlo renormalization-group studies indicate that there is only one transition which is Ising-like. The roles of string and vortex excitations in the phase transition are discussed by an energy-entropy argument and are found to be consistent with Monte Carlo quench results.