Integral representation of the Ising model

TL;DR

This paper presents a method to express the 2D Ising model's partition function as a finite-dimensional integral, highlighting computational challenges due to the large number of coupling constants needed.

Contribution

It introduces an integral representation of the 2D Ising model's partition function that reduces the sum over spins to a finite integral, with insights into computational limitations.

Findings

Partition function expressed as finite integral

Numerical integration required for evaluation

Memory constraints limit the number of spin variables

Abstract

The partition function of the 2D Ising model coupled to an external magnetic field is studied. We show that the sum over the spin variables can be reduced to an integration over a finite number of variables. This integration must be performed numerically. But in order to reduce the partition function we must introduce as many different coupling constants as spin variables. The total memory that we need in order to store these coupling constants imposed important restrictions on the number of spin variables.