Duality in Long-Range Ising Ferromagnets

Abstract

It is proved that for a system of spins $σ_i = \pm 1$ having an interaction energy $-\sum K_{ij} σ_i σ_j $ with all the $K_{ij}$ strictly positive,one can construct a dual formulation by associating a dual spin $S_{ijk} = \pm 1$ to each triplet of distinct sites $i,j$ and $k$. The dual interaction energy reads $-\sum _{(ij)} D_{ij} \prod _{k \neq i,j} S_{ijk}$ with $tanh(K_{ij})\ = \ exp(-2D_{ij})$, and it is invariant under local symmetries. We discuss the gauge-fixing procedure, identities relating averages of order and disorder variables and representations of various quantities as integrals over Grassmann variables. The relevance of these results for Polyakov's approach of the 3D Ising model is briefly discussed.