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

TL;DR

This paper introduces a new duality relation for the two-dimensional Discrete Gaussian SOS model on a torus, revealing a temperature inversion and properties of dual potentials, including a self-dual case with explicit correlation calculations.

Contribution

It constructs a novel duality for 2D Discrete Gaussian models on a torus, based on a 1D duality and the Chinese remainder theorem, extending understanding of potential interactions and correlations.

Findings

Duality relates potentials via temperature inversion $ ilde{eta}=rac{ extpi^2}{eta}$.

In the thermodynamic limit, the dual potential decays as an inverse square law with quadrupolar angular dependence.

At the self-dual temperature, height-height correlations diverge logarithmically and are explicitly calculable.

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 $(v,\widetilde{v})$ of mutually dual potentials and to a temperature inversion according to $\widetilde{\beta}=\pi^2/\beta$. When $v(\mathbf{r})$ is isotropic, duality renders an anisotropic $\widetilde{v}$. 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.