Mapping all classical spin models to a lattice gauge theory

Abstract

In our recent work (De las Cuevas et al 2009 Phys. Rev. Lett. 102 230502), we showed that the partition function of all classical spin models, including all discrete standard statistical models and all Abelian discrete lattice gauge theories (LGTs), can be expressed as a special instance of the partition function of a four-dimensional pure LGT with gauge group ℤ2 (4D ℤ2 LGT). This provides a unification of models with apparently very different features into a single complete model. The result uses an equality between the Hamilton function of any classical spin model and the Hamilton function of a model with all possible k-body Ising-type interactions, for all k, which we also prove. Here, we elaborate on the proof of the result, and we illustrate it by computing quantities of a specific model as a function of the partition function of the 4D ℤ2 LGT. The result also allows one to establish a new method to compute the meanfield theory of ℤ2 LGTs with d ≥ 4, and to show that computing the partition function of the 4D ℤ2 LGT is computationally hard (#P hard). The proof uses techniques from quantum information. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.