 Villain model

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.