Applications of an exact duality-decimation transformation for two-dimensional spin systems on a square lattice

An exact duality-decimation transformation is derived for a large class of two-dimensional spin systems on a square lattice. It maps a model with interactions in every second square (checkerboard interactions) onto a model with interactions in every square and vice versa. The transformation is applicable to Ising, Potts, solid-on-solid (SOS), and x-y models. It may be used to reformulate the site percolation problem in terms of the s→1 limit of a Potts model with four-spin interactions in every elementary square. The transformation maps the van Beijeren SOS model, which is equivalent to the F model, onto an x-y model with checkerboard interactions. Thus one has an exactly soluble x-y model, which, however, bears little resemblance to the ordinary x-y model. The transformation may be combined with a bond-shifting approximation to obtain a real-space renormalization scheme which in some respects represents an improvement over the Migdal-Kadanoff formula. No nearest-neighbor bonds are shifted. Better values of the exponent ν are obtained for the Ising and Potts models. Apart from exponentially vanishing corrections, the renormalization transformation for SOS and x-y models has a surface of fixed points corresponding to generalizations of the usual discrete Gaussian and Villain models.