Blume-Capel mean-field transition at zero single-ion anisotropy ($k_B{T}_c=2zJ/3,S\sim \pm \sqrt{8/3}(-t{)}^{1/2}$)

For spins $s_i\in \{-1,0,1\}$ with ferromagnetic nearest-neighbour exchange and no single-ion anisotropy, a homogeneous mean-field approximation givesThe 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 $k_B{T}_c=2zJ/3$ 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.