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 
(4D 
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 
LGT. The result also allows one to establish a new method to compute the mean-field theory of 
LGTs with d⩾4, and to show that computing the partition function of the 4D 
LGT is computationally hard (#P hard). The proof uses techniques from quantum information.
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GENERAL SCIENTIFIC SUMMARY Introduction and background. The keystone of a statistical mechanical system is its partition function, for all its thermodynamical quantities can be derived from it, like the free energy, magnetization, etc. However, knowledge of the partition function of a particular system does not necessarily yield any insight into other systems. Via a detour to quantum information science, it was recently discovered that the partition function of a large class of models can be re-expressed as the partition function of the Ising model with magnetic fields in two dimensions (and larger), the complete model. The interaction strengths of the complete model thereby determine whether its partition function becomes, say, that of an eight-dimensional Ising model, or that of a model with many-body interactions, for instance.
Main results. We have widened the scope of this result by including a very different family of models: lattice gauge theories. In their simplest form, these are statistical mechanical models, whose salient feature is to have local symmetries instead of global symmetries. This new richness has enabled us to prove that the partition function of any spin model can be recast into that of a lattice gauge theory in four dimensions and with the simplest gauge group, Z2.
Wider implications. The result is at first sight rather surprising, since it `unifies' notably different models – models in different universality classes, e.g. with different dimensions, different types of interactions, and also with local and global symmetries. Our constructive proof has given us some insight into how this is at all possible.
Figure. The partition function of a specific model, in this case the two-dimensional Ising model, is recovered from the partition function of a four-dimensional Z2 lattice gauge theory with the appropriate coupling strengths (here indicated by colors and arrows).