Past exam of the quantum condensed matter course of the University of Cambridge / 2019 / Section B / 6 / d

Close to the critical temperature on the ordered side, both $m$ and $m/t$ are small. Expanding the hyperbolic tangent givesFor a nonzero solution, divide by $m$ and retain the cubic-order terms:This mean-field onset of two-state ferromagnetism has mean-field critical exponent $1/2$: the curve reaches zero continuously with a square-root cusp below $t=1$, and remains exactly zero for all $t\ge 1$. The factor $t$ is what the cubic truncation gives; higher orders also affect the coefficient at relative order $1-t$, so the robust leading asymptote is $m\sim \sqrt{3(1-t)}$.