The high temperature expansion of the classical $XYZ$ chain

TL;DR

This paper derives a high-temperature expansion of the Helmholtz free energy for the classical XYZ chain with anisotropy and magnetic field, validating it against numerical solutions and analyzing thermodynamic properties.

Contribution

The paper provides a detailed $eta$-expansion up to order 12 for the classical XYZ model, including anisotropy and magnetic field effects, extending previous analytical results.

Findings

Expansion valid for intermediate temperatures around $kT/J_x \\approx 0.5$

Classical results approximate quantum thermodynamics up to $kT/J \\approx 0.8$ with 2.5% error

Ferromagnetic and antiferromagnetic chains share the same free energy without magnetic field

Abstract

We present the $\beta$-expansion of the Helmholtz free energy of the classical $XYZ$ model, with a single-ion anisotropy term and in the presence of an external magnetic field, up to order $\beta^{12}$. We compare our results to the numerical solution of Joyce's [Phys. Rev. Lett. 19, 581 (1967)] expression for the thermodynamics of the $XXZ$ classical model, with neither single-ion anisotropy term nor external magnetic field. This comparison shows that the derived analytical expansion is valid for intermediate temperatures such as $kT/J_x \approx 0.5$. We show that the specific heat and magnetic susceptibility of the spin-2 antiferromagnetic chain can be approximated by their respective classical results, up to $kT/J \approx 0.8$, within an error of 2.5%. In the absence of an external magnetic field, the ferromagnetic and antiferromagnetic chains have the same classical Helmholtz free energy. We show how this two types of media react to the presence of an external magnetic field.