Some Critical Properties of the Heisenberg Model

Abstract

A detailed series-extrapolation investigation of some critical properties ($T\to T_c+$) of the Heisenberg model is presented. Systematic use is made of ratio and Padé-approximant techniques, some of the latter being new. Attention is confined to those nearest-neighbor and order-two equivalent-neighbor models which are based on the simple cubic, body-centered cubic, and face-centered cubic lattices. For the face-centered cubic nearest-neighbor classical Heisenberg model, the following estimates are obtained for familiar critical exponents: $\alpha =0, -\frac{1}{8}<~{\alpha }_s<~-\frac{1}{16}$, $\gamma =1.375\mathrm{}\pm 0.002$, $0.6875<~\nu <~0.7125$, $0<~\eta <~0.07$. Estimates for $\gamma$ are also obtained for the other nearest-neighbor and equivalent-neighbor classical Heisenberg models, and it is conjectured that $\gamma =1\mathrm{}\frac{3}{8}$ may be the exact result in all the cases considered. For the face-centered cubic order-two equivalent-neighbor spin-½ Heisenberg ferromagnet, it is estimated that $\gamma =1.3744\mathrm{}\pm 0.0008$. A similar but less precise result is found for the corresponding simple cubic model, and the spin-½ nearest-neighbor susceptibility series are briefly examined. It is suggested that consideration should be given to the possibility that for ferromagnetic Heisenberg interactions $\gamma$ has the same value $1\frac{3}{8}$ irrespective of spin and the particular three-dimensional nearest-neighbor or finite-order equivalent-neighbor model considered. Other quantities estimated include various Curie points and singularity amplitudes and the critical values of certain thermodynamic functions. Some attention is given to the simple classical Heisenberg antiferromagnet.