Correlation functions in XY models and step free energies in roughening models

A duality relation derived by Jos\'e, Kadanoff, Kirkpatrick, and Nelson and by Knops is exploited to calculate properties of XY and roughening models from known properties of the dual models. It is shown that the correlation length in XY models is exactly given by ξ^-1= ̃ \~β ̃ \~f, where ̃ \~f is the free energy per unit length of a step and ̃ \~β is the inverse temperature in the corresponding roughening model. Similarly, the exponent η below Tc in an XY model is given by ${η}={lim}_{|{{→}}{r}|{→}{∞}}{{ ̃ \~{}{}}{{β}}{{ ̃ \~{}{}}{F}}₁({{→}}{r})}{ln|{{→}}{r}|}$, where ${{ ̃ \~{}{}}{F}}₁({{→}}{r})$ is the free energy associated with two screw dislocations of unit strength and opposite sign separated by the distance ${{→}}{r}$ in the corresponding roughening model.