Effects of nearest-neighbor four-spin correlation upon the critical properties of the spin-1/2 Heisenberg ferromagnet

Abstract

A spin-1/2 fcc isotropic Heisenberg ferromagnet is treated by the cluster-variation method in which the nearest-neighbor pair, triangle, and tetrahedral correlations are taken into account. The Helmholtz free energy, which is expressed in terms of eight quantities, $〈S^z〉$, $〈S_1^z{S}_2^z〉$, $〈S_1^{-}{S}_2^{+}〉$, $〈S_1^z{S}_2^z{S}_3^z〉$, $〈S_1^{-}{S}_2^{+}{S}_3^z〉$, $〈S_1^z{S}_2^z{S}_3^z{S}_4^z〉$, $〈S_1^{-}{S}_2^{+}{S}_3^z{S}_4^z〉$, and $〈S_1^{-}{S}_2^{-}{S}_3^{+}{S}_4^{+}〉$, is minimized with respect to those expectation values. Above the Curie temperature three of these quantities vanish and there are only two linearly independent variables. Below $T_c$ the eight variables satisfy nontrivial nonlinear transcendental equations. The critical data obtained are $〈S^z〉=〈S_1^z{S}_2^z{S}_3^z〉=〈S_1^{+}{S}_2^{-}{S}_3^z〉=0\mathrm{}$, $2〈S_1^{+}{S}_2^{-}〉=4\mathrm{}〈S_1^z{S}_2^z〉=\frac{1}{5}$, $6〈S_1^{+}{S}_2^{+}{S}_3^{-}{S}_4^{-}〉=24\mathrm{}〈S_1^{+}{S}_2^{-}{S}_3^z{S}_4^z〉=16\mathrm{}〈S_1^z{S}_2^z{S}_3^z{S}_4^z〉=0.081\mathrm{}802$, and $T_c=0.6735*T_c$ (Weiss). It is found that ${〈S^z〉}^3$ is more linear than ${〈S^z〉}^2$ over a wide range of temperature below $T_c$.