Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 355 / 3 / d

Write the free energy asWhen $A_H<A_H^{\ast }$, $B_2>0$ and the global minimum on $\varphi \ge 0$ is ${\varphi }_{\ast }=0$, corresponding to an unbound stack. When $A_H>A_H^{\ast }$, $B_2<0$, and minimization givesUsing $B_2\sim -r(A_H-A_H^{\ast })$ yieldsSince $d_{\ast }=\delta /{\varphi }_{\ast }$,so the continuous membrane unbinding transition has exponent