= Villain model
{c}
{title2=$Z_V=\int\prod_x\frac{d\theta_x}{2\pi}\prod_e\sum_{n_e\in\mathbb Z}\frac{e^{-n_e^2/(2\beta)+in_e\Delta_e\theta}}{\sqrt{2\pi\beta}}$}
A Villain model replaces the cosine <Boltzmann factor> of the <XY model> by a periodic <Gaussian> weight. The <Poisson summation formula> makes its bond weight equal to $\sum_{p\in\mathbb Z}\exp[-\beta(\psi-2\pi p)^2/2]$. The periodic Gaussian retains integer winding defects and permits an explicit separation into <spin waves> and <vortices> in two dimensions. Its replacement of the cosine model is an approximation; subsequent transformations within the Villain model can be exact.
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