= Blume-Capel mean-field transition at zero single-ion anisotropy
{c}
{title2=$k_BT_c=2zJ/3,\quad S\sim\pm\sqrt{8/3}\,(-t)^{1/2}$}
For spins $s_i\in\{-1,0,1\}$ with ferromagnetic nearest-neighbour exchange and no single-ion anisotropy, a homogeneous <mean-field approximation> gives
$$
\frac{F_{\rm MF}}N=\frac{zJ}{2}S^2-\frac1\beta\log[1+2\cosh\beta(zJS+B)].
$$
The <self-consistency equation> is $S=2\sinh h/(1+2\cosh h)$ with $h=\beta(zJS+B)$. Its small-$h$ expansion is $2h/3-h^3/9+O(h^5)$. Hence $k_BT_c=2zJ/3$ and, with $t=T/T_c-1$, the nonzero branches satisfy $S^2=(8/3)(-t)+O(t^2)$. The <critical exponent> is the mean-field value $1/2$, not an exact two-dimensional exponent. At zero source the finite-system equilibrium magnetization is zero by <spin inversion symmetry>; the broken branches describe the thermodynamic limit with a phase selected by a vanishing source.
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