Extended Scaling for the high dimension and square lattice Ising Ferromagnets

TL;DR

This paper introduces an extended scaling approach that accurately describes the temperature dependence of susceptibility and correlation length in high-dimensional and two-dimensional square lattice Ising ferromagnets across all temperatures, challenging the notion of a crossover temperature.

Contribution

It proposes a novel extended scaling method that provides accurate, continuous descriptions of critical properties in finite-dimensional Ising models without crossover temperature assumptions.

Findings

Extended scaling accurately models susceptibility and correlation length.

The approach applies to both high and two-dimensional Ising ferromagnets.

No crossover temperature is needed for mean-field-like behavior in finite dimensions.

Abstract

In the high dimension (mean field) limit the susceptibility and the second moment correlation length of the Ising ferromagnet depend on temperature as chi(T)=tau^{-1} and xi(T)=T^{-1/2}tau^{-1/2} exactly over the entire temperature range above the critical temperature T_c, with the scaling variable tau=(T-T_c)/T. For finite dimension ferromagnets temperature dependent effective exponents can be defined over all T using the same expressions. For the canonical two dimensional square lattice Ising ferromagnet it is shown that compact "extended scaling" expressions analogous to the high dimensional limit forms give accurate approximations to the true temperature dependencies, again over the entire temperature range from T_c to infinity. Within this approach there is no cross-over temperature in finite dimensions above which mean-field-like behavior sets in.