An extended scaling analysis of the S=1/2 Ising ferromagnet on the simple cubic lattice

TL;DR

This paper introduces an extended scaling approach for analyzing continuous phase transitions, demonstrating its effectiveness on the 3D Ising model and showing critical behavior extends over a wider temperature range than traditionally assumed.

Contribution

The paper develops and applies an extended scaling method based on high temperature series expansions, broadening the applicability of Renormalization Group analysis.

Findings

Extended scaling captures critical behavior from Tc to infinite temperature.

Proper scaling variables improve data analysis over wide temperature ranges.

Numerical analysis confirms the extended scaling approach's validity.

Abstract

It is often assumed that for treating numerical (or experimental) data on continuous transitions the formal analysis derived from the Renormalization Group Theory can only be applied over a narrow temperature range, the "critical region"; outside this region correction terms proliferate rendering attempts to apply the formalism hopeless. This pessimistic conclusion follows largely from a choice of scaling variables and scaling expressions which is traditional but which is very inefficient for data covering wide temperature ranges. An alternative "extended caling" approach can be made where the choice of scaling variables and scaling expressions is rationalized in the light of well established high temperature series expansion developments. We present the extended scaling approach in detail, and outline the numerical technique used to study the 3d Ising model. After a discussion of the exact expressions for the historic 1d Ising spin chain model as an illustration, an exhaustive analysis of high quality numerical data on the canonical simple cubic lattice 3d Ising model is given. It is shown that in both models, with appropriate scaling variables and scaling expressions (in which leading correction terms are taken into account where necessary), critical behavior extends from Tc up to infinite temperature.