Duality in Long-Range Ising Ferromagnets

TL;DR

This paper establishes a dual formulation for long-range ferromagnetic Ising models with positive interactions, revealing symmetries and gauge structures that deepen understanding of their phase behavior.

Contribution

It introduces a dual spin representation for long-range Ising models, linking interaction parameters via hyperbolic tangent relations and exploring gauge invariances and integral representations.

Findings

Dual spin formulation with triplet association

Invariant under local symmetries

Connections to Grassmann variable integrals

Abstract

It is proved that for a system of spins $\sigma _i = \pm 1$ having an interaction energy $-\sum K_{ij} \sigma _i \sigma _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.