New duality relation for the Discrete Gaussian SOS model on a torus
Résumé
We construct a new duality for two-dimensional Discrete Gaussian (DG) 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 N M 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, \widetilde{v}) of mutually dual potentials and to a temperature inversion according to \widetilde{β} = π^2 /β. When the potential v(r) is isotropic, \widetilde{v} inherits an anisotropy from the mapping. This is the case for the potential \widetilde{v} that is dual to an isotropic nearest-neighbor potential v. In the thermodynamic limit the latter dual potential is shown to decay with distance according to an inverse square law with a simple angular dependence. There is a single self-dual pair of potentials (v* , \widetilde{v*}). At the self-dual temperature β* = \widetilde{β*} = π the height-height correlation can be calculated explicitly; it is anisotropic and diverges logarithmically with distance.
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