Smooth and global Ising universal scaling functions

TL;DR

This paper introduces a polynomial-based method to approximate universal scaling functions for the 2D Ising model in a field, accurately capturing key singularities and providing computational tools.

Contribution

It presents a novel polynomial series approach using parametric coordinates to approximate Ising model scaling functions with high accuracy.

Findings

Converges exponentially with series order, achieving up to seven digits of accuracy.

Provides computational implementations in Python and Mathematica.

Successfully captures essential singularities like Langer and Yang-Lee edges.

Abstract

We describe a method for approximating the universal scaling functions for the Ising model in a field. By making use of parametric coordinates, the free energy scaling function has a polynomial series everywhere. Its form is taken to be a sum of the simplest functions that contain the singularities which must be present: the Langer essential singularity and the Yang--Lee edge singularity. Requiring that the function match series expansions in the low- and high-temperature zero-field limits fixes the parametric coordinate transformation. For the two-dimensional Ising model, we show that this procedure converges exponentially with the order to which the series are matched, up to seven digits of accuracy. To facilitate use, we provide Python and Mathematica implementations of the code at both lowest order (three digit) and high accuracy.