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

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 $[$ $\mathcal{H}={\sum }_{i=1}^N\{-J{\mathbf{S}}_i{\mathbf{S}}_{i+1}-(h/S)S_i^z\}$ $(J>0),$ $]$ 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 $S.$

16 pages (revtex 2.0) + 6 PS figures upon request

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