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''. Each interaction is associated with a linear combination of variables; these are summarized in a matrix $J$. A Gibbs factor is associated to each variable (one-body term) and to each interaction. Then 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. If the matrix $J$ is interpreted as a vector representation of a matroid, duality exchanges the matroid with its dual. We discuss some physical examples. The main idea is to generalize known models up to eventually include randomness into the pattern of interaction. We introduce and study a random Gaussian , a randomPotts-like model, and a random variant of discrete scalar QED. Although the classical procedure \em 'a la Kramers and Wannier does not extend in a natural way to such a wider class of systems, our weaker procedure applies to these models, too. We shortly describe the consequence of duality for each example.
|Titolo:||General duality for abelian-group-valued statistical- mechanics models|
|Parole Chiave:||duality (mathematics) ; free energy ; gauge field theory ; Potts model ; quantum electrodynamics ; statistical mechanics|
|Settore Scientifico Disciplinare:||Settore FIS/02 - Fisica Teorica, Modelli e Metodi Matematici|
|Data di pubblicazione:||2004|
|Digital Object Identifier (DOI):||10.1088/0305-4470/37/30/002|
|Appare nelle tipologie:||01 - Articolo su periodico|