Duality in Long-Range Ising Ferromagnets

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.Comment: 16 pp., UIOWA-91-2

Remarks Concerning Polyakov's Conjecture for the 3D Ising Model and the Hierarchical Approximation

We consider the possibility of using the hierarchical approximation to understand the continuum limit of a reformulation of the 3D Ising model initiated by Polyakov. We introduce several new formulations of the hierarchical model using dual or fermionic variables. We discuss several aspects of the renormalization group transformation in terms of these new variables. We mention a reformulation of the model closely related to string models proposed by Zabrodin.Comment: 7 pp., UIOWA-91-2

Quantum Simulating Lattice Gauge Theories with Optical Lattices

Presentation from Near-term Applications of Quantum Computing, Fermilab, 06-07 Dec 201