An algorithmic proof for the completeness of two-dimensional Ising model

TL;DR

This paper proves that the two-dimensional Ising model can represent the partition functions of any lattice model through an algorithmic transformation, making the universality of the Ising model more accessible without quantum formalism.

Contribution

It provides a purely classical, algorithmic proof of the Ising model's completeness, avoiding quantum techniques and enabling transformations between models.

Findings

Partition functions of any lattice model can be transformed into a 2D Ising model.

The transformation is algorithmic and involves simple graphical steps.

The resulting Ising model has complex couplings and adjusted local fields.

Abstract

We show that the two dimensional Ising model is complete, in the sense that the partition function of any lattice model on any graph is equal to the partition function of the 2D Ising model with complex coupling. The latter model has all its spin-spin coupling equal to i\pi/4 and all the parameters of the original model are contained in the local magnetic fields of the Ising model. This result has already been derived by using techniques from quantum information theory and by exploiting the universality of cluster states. Here we do not use the quantum formalism and hence make the completeness result accessible to a wide audience. Furthermore our method has the advantage of being algorithmic in nature so that by following a set of simple graphical transformations, one is able to transform any discrete lattice model to an Ising model defined on a (polynomially) larger 2D lattice.