Critical temperatures of Heisenberg magnets by second-order Green's-function theory: Application toS=∞cubic lattices and quasi-one-dimensional systems

The critical temperature $T_c$ of a Heisenberg magnet is estimated as the point where the second moment $〈{\omega }_{q_0}^2〉$, as approximated by a second-order Green's-function decoupling, goes to zero at the wave vector of magnetic order $q_0$. This condition implies a relation among the short-range correlations which is obeyed to within 1% or better for $S=\infty$ cubic lattices, using the series results for spin correlations at $T_c$. It further leads to a relation among intrachain ($J$) and interchain ($J^{\prime }$) exchange constants and $T_c$ for classical linear chains with weak interchain coupling which is of the same form as given by the random-phase-approximation first-order Green's-function theory. The value of $J^{\prime }$ predicted in terms of $J$ and $T_c$ is compared with neutron measurements in CsMn${\mathrm{Cl}}_3$·2${\mathrm{H}}_2$O.