Scaling theory for classical anisotropic Heisenberg chain

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Abstract
The classical anisotropic Heisenberg chain (with Jperpendicular to /J/sub ///= alpha ) is treated by a simple scaling theory valid in the low-temperature limit (K identical to J/sub ////KBT>>1) for any alpha between 0 (Ising) and 1 (isotropic case). The correlation lengths are given in terms of universal single-variable scaling functions Phi /sub //, perpendicular to /: 1/ xi /sub //, perpendicular to /=(cosh-1(1/ alpha )) Phi /sub //, perpendicular to /(K(1- alpha 2)12/+F( alpha )). The term F( alpha ), given in the text, is negligible except when alpha approximately 0. The asymptotic forms of the functions are: Phi /sub ///(x) approximately Phi perpendicular to (x) approximately A/x for x<>1 (where A, B and C are constants), giving the characteristic 'Heisenberg-like' and 'Ising-like' behaviours occurring on either side of the crossover which takes place at x approximately 1. Exact critical and crossover exponents are deduced.