 Classical Heisenberg Langevin self-consistency

The single-site average in molecular field $z_{c}JS$ is $S=\coth x-1/x$, where $x=\beta z_{c}JS$. Thus $\tau x=\coth x-1/x$ with $\tau =k_{B}T/(z_{c}J)$. The small-$x$ slope $1/3$ gives $k_B{T}_c^{\mathrm{MF}}=z_{c}J/3$. On a square lattice $z_c=4$. This finite mean-field transition does not imply actual positive-temperature magnetization for the ideal two-dimensional short-range isotropic model, where the Mermin-Wagner theorem applies.