General duality for abelian-group-valued statistical-mechanics models

TL;DR

This paper introduces a duality transformation for a broad class of abelian-group-valued statistical-mechanics models, encompassing spin, gauge, ordered, and disordered systems, extending classical duality concepts to include randomness.

Contribution

It develops a general duality framework for abelian-group-valued models, including random and disordered systems, and explores its implications for various physical examples.

Findings

Duality exchanges group with its dual and Fourier transforms of Gibbs factors.

High and low couplings are interchanged under the duality transformation.

The framework applies to models like Gaussian, Potts-like, and scalar QED with randomness.

Abstract

We introduce a general class of statistical-mechanics models, taking values in an abelian group, which includes examples of both spin and gauge models, both ordered and disordered. The model is described by a set of ``variables'' and a set of ``interactions''. A Gibbs factor is associated to each variable and to each interaction. We introduce a duality transformation for systems in this class. The duality exchanges the abelian group with its dual, the Gibbs factors with their Fourier transforms, and the interactions with the variables. High (low) couplings in the interaction terms are mapped into low (high) couplings in the one-body terms. The idea is that our class of systems extends the one for which the classical procedure 'a la Kramers and Wannier holds, up to include randomness into the pattern of interaction. We introduce and study some physical examples: a random Gaussian Model, a random Potts-like model, and a random variant of discrete scalar QED. We shortly describe the consequence of duality for each example.