In order to find the correct vortex lattice configuration (i.e. ground state) in Ginzburg-Landau theory (or the Abelian Higgs Model), it is standard practice to minimize the beta parameter:
$\beta=\frac{\langle |\phi|^{4}\rangle}{\langle |\phi|^{2}\rangle}$.
What is the difference between minimizing $\beta$ and minimizing $\langle |\phi|^{4}\rangle$? Why are they not equivalent?