The Free Energy and the Scaling Function of the Ferromagnetic Heisenberg Chain in a Magnetic Field

TL;DR

This paper investigates the low-temperature nonlinear and linear susceptibilities of ferromagnetic Heisenberg chains, revealing divergence behaviors and a universal scaling function for magnetization across different spin values.

Contribution

It derives a universal scaling relation and function for the magnetization of the Heisenberg ferromagnet applicable to all spin values, extending previous specific cases.

Findings

Nonlinear susceptibilities diverge as T^{-6}.

Linear susceptibilities diverge as T^{-2}.

A universal scaling function F(x) is identified for all spins.

Abstract

A nonlinear susceptibilities (the third derivative of a magnetization $m_S$ by a magnetic field $h$ ) of the $S$=1/2 ferromagnetic Heisenberg chain and the classical Heisenberg chain are calculated at low temperatures $T.$ In both chains the nonlinear susceptibilities diverge as $T^{-6}$ and a linear susceptibilities diverge as $T^{-2}.$ The arbitrary spin $S$ Heisenberg ferromagnet $[$ ${\cal H} = \sum_{i=1}^{N} \{ - J{\bf S}_{i} {\bf S}_{i+1} - (h/S) S_{i}^{z} \}$ $(J>0),$ $]$ has a scaling relation between $m_S,$ $h$ and $T:$ $m_S(T,h) = F( S^2 Jh/T^2).$ The scaling function $F(x)$=(2$x$/3)-(44$x^{3}$/135) + O($x^{5}$) is common to all values of spin $S.$