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

F Cornu (1) , H J Hilhorst , M Bauer (2)

1 LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
2 IPHT - Institut de Physique Théorique - UMR CNRS 3681

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, tv) of mutually dual potentials and to a temperature inversion according to tβ = π^2 /β. When the potential v(r) is isotropic, tv inherits an anisotropy from the mapping. This is the case for the potential tv 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* , tv*). At the self-dual temperature β* = tβ* = π the height-height correlation can be calculated explicitly; it is anisotropic and diverges logarithmically with distance.

Mots clés: two-dimensional duality; SoS Discrete Gaussian model; Chinese remainder theorem

Domaines: Physique [physics]

F Cornu, H J Hilhorst, M Bauer. New duality relation for the Discrete Gaussian SOS model on a torus. 2002. ⟨hal-03861620v1⟩