DUALITY IN LONG-RANGE ISING-FERROMAGNETS

It is proved that for a system of spins sigma(i) = +/- 1 having an interaction energy -SIGMAK(ij)sigma(i)sigma(j) with all the K(ij) strictly positive, one can construct a dual formulation by associating a dual spin S(ij)k = +/- 1 to each triplet of distinct sites i, j, and k. The dual interaction energy reads -SIGMA(ij)D(ij)PI(k not-equal i,j)S(ijk) with tanh(K(ij)) = exp(-2D(ij)), and it is invariant under local symmetries. The gauge-fixing procedure, identities relating averages of order and disorder variables, and representations of various quantities as integrals over Grassmann variables are discussed. The relevance of these results for Polyakov's approach of the 3-D Ising model is briefly discussed.