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Abstract
We study the crossover between two- and three-dimensional behaviors of the Villain form of the XY spin model on a three-dimensional lattice with anisotropic couplings ${\mathit{J}}_{1}$=${\mathit{J}}_{2}$=J, ${\mathit{J}}_{3}$=\ensuremath{\alpha}J (0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}1). The \ensuremath{\alpha} dependence of various quantities is examined by means of duality transformations, Migdal renormalization group, and Monte Carlo simulations. For the specific heat, a crossover into the Kosterlitz-Thouless behavior takes place around \ensuremath{\alpha}\ensuremath{\simeq}0.2, which corresponds to ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{cos}}$\ensuremath{\simeq}0.015 in the cosine form of the XY model. Some implications for models of high-${\mathit{T}}_{\mathit{c}}$ superconductivity are discussed.
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