Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 50 / 1 / f

At a tricritical point, both $r$ and $u$ vanish, while $v>0$ stabilizes the Landau free energy. It separates ordinary continuous transitions for $u>0$ from first-order coexistence for $u<0$. The zero-field order-parameter jump shrinks to zero as $u\uparrow 0$, and the tricritical wings terminate there in the full $(r,u,h)$ diagram. For $r<0$, $u=0$, minimizing gives $M^4=-r/v$ andTwo thermal derivatives give the heat-capacity critical exponent $\alpha =1/2$, compared with the finite mean-field heat-capacity jump, $\alpha =0$, at an ordinary quartic transition. The remaining tricritical mean-field critical exponents are $(\beta ,\gamma ,\delta )=(1/4,1,5)$, and the mean-field correlation-length critical exponent is $\nu =1/2$. To observe tricritical scaling one must tune the quartic scaling field as well as the thermal field: a generic path with fixed $u>0$ instead has ordinary critical behavior. Its upper critical dimension is three, as the multicritical Ginzburg ratio shows.