Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 303 / 1 / f

On a tricritical thermal path, the quartic scaling variable $X$ tends to zero, and at zero field $Y=0$. On the ordered side the tricritical crossover scaling function tends to ${\Phi }_{-}(0,0)=-1/3$. Consequentlyprovided the quadratic coefficient has a nonzero thermal slope and the sextic coefficient has a finite positive limit. Since the heat-capacity critical exponent satisfies $f_{\mathrm{s}}\sim |T-T_c{|}^{2-\alpha }$,Equivalently, twice differentiating this ordered-side free-energy density produces the $(T_c-T{)}^{-1/2}$ singularity. The scaling path and branch qualifications are the same as for the direct Landau theory calculation: a nonzero fixed positive quartic coefficient would instead send $X\to \infty$ and yield the ordinary transition of part (e).