Unifying all classical spin models in a Lattice Gauge Theory

TL;DR

This paper demonstrates that all classical spin models and abelian discrete lattice gauge theories can be represented within a single 4D Z_2 lattice gauge theory, unifying diverse models and revealing their computational complexity.

Contribution

It establishes a universal framework by expressing various classical spin models as special cases of the 4D Z_2 LGT, linking different models physically and computationally.

Findings

Partition functions of all classical spin models are reducible to the 4D Z_2 LGT

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

The 4D Z_2 LGT is approximately complete for abelian continuous models

Abstract

We show 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 the 4D Z_2 LGT. In this way, all classical spin models with apparently very different features are unified in a single complete model, and a physical relation between all models is established. As applications of this result, we present a new method to do mean field theory for abelian discrete LGTs with d>3, and we show that the computation of the partition function of the 4D Z_2 LGT is a computationally hard (#P-hard) problem. We also extend our results to abelian continuous models, where we show the approximate completeness of the 4D Z_2 LGT. All results are proven using quantum information techniques.