New duality relation for the Discrete Gaussian SOS model on a torus

Abstract

We construct a new duality for two-dimensional Discrete Gaussian models. It is based on a known one-dimensional duality and on a mapping, implied by the Chinese remainder theorem, between the sites of an $N\times M$ torus and those of a ring of $NM$ sites. The duality holds for an arbitrary translation invariant interaction potential $v(\mathbf{r})$ between the height variables on the torus. It leads to pairs $(v,\tilde{v})$ of mutually dual potentials and to a temperature inversion according to $\tilde{\beta }={\pi }^2/\beta$. When $v(\mathbf{r})$ is isotropic, duality renders an anisotropic $\tilde{v}$. This is the case, in particular, for the potential that is dual to an isotropic nearest-neighbor potential. In the thermodynamic limit this dual potential is shown to decay with distance according to an inverse square law with a quadrupolar angular dependence. There is a single pair of self-dual potentials $v^{\star }=\widetilde{v^{\star }}$. At the self-dual temperature ${\beta }^{\star }=\widetilde{{\beta }^{\star }}=\pi$ the height-height correlation can be calculated explicitly; it is anisotropic and diverges logarithmically with distance.

26 pages, 2 figures