Mapping all classical spin models to a lattice gauge theory

TL;DR

This paper demonstrates that all classical spin models and Abelian lattice gauge theories can be represented within a single 4D Z_2 lattice gauge theory, unifying diverse models and enabling new computational approaches.

Contribution

It proves the equivalence of classical spin models and 4D Z_2 lattice gauge theories, providing a unified framework and new methods for analysis.

Findings

Partition functions of all classical spin models can be expressed as that of a 4D Z_2 LGT.

The Hamilton function of any classical spin model equals that of a model with all possible k-body Ising interactions.

Computing the partition function of 4D Z_2 LGT is #P hard.

Abstract

In our recent work [Phys. Rev. Lett. 102, 230502 (2009)] 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 4-dimensional pure LGT with gauge group Z_2 (4D Z_2 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 Z_2 LGT. The result also allows one to establish a new method to compute the mean-field theory of Z_2 LGTs with d > 3, and to show that computing the partition function of the 4D Z_2 LGT is computationally hard (#P hard). The proof uses techniques from quantum information.