= Mean-field Ising critical bifurcation
{title2=$M^2\sim3(T_c-T)/T_c,\quad k_BT_c=zJ$}
For a ferromagnetic <Ising model> with <coordination number of a lattice> $z$, its <mean-field self-consistency equation> is $M=\tanh(zJM/(k_BT))$. Put $A=T_c/T$. For $A\leq1$, the only solution is zero. For $A>1$, <strict concavity> of $\tanh(AM)-M$ on $M>0$ gives one positive solution and its spin-reversed partner. Expanding $\tanh(AM)=AM-(AM)^3/3+O(M^5)$ gives $M^2=3(A-1)/A^3+O((A-1)^2)$, proving a continuous onset with <order-parameter critical exponent> $\beta=1/2$. The <Bragg-Williams free energy of the Ising model> has a positive quartic coefficient at $T_c$, so these are the stable ordered minima rather than a first-order jump.
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