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

A nonlinear susceptibilities (the third derivative of a magnetization mS 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 $[$ H = ∑ᵢ₌₁N \ - J Sᵢ Sᵢ₊₁ - (h/S) Sᵢᶻ \ $(J>0),$ $]$ has a scaling relation between mS, h and $T:$ mS(T,h) = F( S² Jh/T²). The scaling function $F(x)$=(2x/3)-(44x³/135) + O(x⁵) is common to all values of spin $S.$