Extended Scaling for Ferromagnets

TL;DR

This paper introduces an extended scaling scheme for ferromagnets that optimizes the representation of thermodynamic observables near critical points, improving data analysis and interpretation.

Contribution

It proposes a systematic rule for choosing scaling variables and non-critical prefactors based on HTSE results, enhancing the analysis of critical behavior in ferromagnets.

Findings

The scheme aligns with standard susceptibility scaling above Tc.

It provides new expressions for specific heat and correlation length.

Validated through high-precision numerical and experimental data.

Abstract

A simple systematic rule, inspired by high-temperature series expansion (HTSE) results, is proposed for optimizing the expression for thermodynamic observables of ferromagnets exhibiting critical behavior at $\Tc$. This ``extended scaling'' scheme leads to a protocol for the choice of scaling variables, $\tau=(T-\Tc)/T$ or $(T^2 - \Tc^2)/T^2$ depending on the observable instead of $(T-\Tc)/\Tc$, and more importantly to temperature dependent non-critical prefactors for each observable. The rule corresponds to scaling of the leading of the reduced susceptibility above $\Tc$ as $\chi_{\rm c}^{*}(T)\sim \tau^{-\gamma}$ in agreement with standard practice with scaling variable $\tau$, and for the leading term of the second-moment correlation length as $\xi_{\rm c}^{*}(T)\sim T^{-1/2}\tau^{-\nu}$. For the specific heat in bipartite lattices the rule gives $C_{\rm c}^{*}(T) \sim T^{-2}[(T^2 -\Tc^2)/T^2]^{-\alpha}$. The latter two expressions are not standard. The scheme can allow for confluent and non-critical correction terms. A stringent test of the extended scaling is made through analyses of high precision numerical and HTSE data, or {\it real} data, on the three-dimensional canonical Ising, XY, and Heisenberg ferromagnets.