Abrikosov's Vortex Lattice (Beta Parameter)

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?

• The answer can be found in Abrikosov's paper: Sov. Phys. JETP 5 1174 (1957). The free energy for the vortex lattice is negative below the critical magnetic field and inversely proportional to $\beta$. This is NOT equivalent to minimizing $\langle |\phi|^{4}\rangle$. – 121 May 1 '18 at 12:12

$F=H_{0}^{2}-\frac{(H_{c}+H_{0})(H_{c}-H_{0})}{\beta(2\chi^{2}-1)}$, where $H_{0}$ is the external magnetic field, $H_{c}\equiv\chi$ is the critical field above which the vortex lattice is unstable and the ground state is the normal vacuum, $\beta_{A}$ is the Abrikosov parameter that determines the lattice structure and $\chi$ is the only free parameter in the theory. The minimum value $\beta$ can take is $1.1596$ for a hexagonal lattice. This assertion is valid for $\chi>\frac{1}{\sqrt{2}}$. Therefore, minimizing $\beta$ gives the correct ground state, it is not equivalent to minimizing $\langle |\phi|^{4}\rangle$.