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

Abstract

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 v ( r ) between the height variables on the torus. It leads to pairs ( v , v ˜ ) of mutually dual potentials and to a temperature inversion according to β ˜ = π 2 / β . When v ( r ) is isotropic, duality renders an anisotropic 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 ⋆ = v ⋆ ˜ . At the self-dual temperature β ⋆ = β ⋆ ˜ = π the height–height correlation can be calculated explicitly; it is anisotropic and diverges logarithmically with distance.