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 × M torus and those of a ring of NM sites. The duality holds for an arbitrary translation-invariant interaction potential between the height variables on the torus. It leads to pairs of mutually dual potentials and to a temperature inversion according to . When is isotropic, duality renders an anisotropic . 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 . At the self-dual temperature the height–height correlation can be calculated explicitly; it is anisotropic and diverges logarithmically with distance.