# Spatially uniform VEV for gauged Ginzburg-Landau functional

$$F = \int dV \left \{\alpha |\psi|^2 + \frac{\beta}{2}|\psi|^4 + \frac{1}{2m^*} \mid (\frac{\hbar}{i}\nabla - \frac{e^*}{c}A)\psi \mid^2 + \frac{h^2}{8\pi}\right \}$$
I will write $\psi = \eta e^{i \phi}$ in the next lines, with both $\eta$ and $\phi$ real functions.
Now if the gauge field $A$ is null the absolute minimum of the functional has surely $\eta = const$. But if the gauge field is not null (it can be fixed from the outside or determined by the superconducting currets) it doesn't seem obvious to me that the VEV should be spatially uniform. Substitution of $\psi = \eta e^{i \phi}$ into the functional gives a contribute
$$\int dV \left \{ |\nabla \eta |^2 + 2 A \cdot \nabla \eta \right \}$$
for spatially varing $\eta$. Determination of the absolute minimum is pretty important as one will then expand the field $\psi$ around this. How can I see absolute mimum should correspond to $\eta = const$ even in the case of non null gauge field?
• In general $\eta$ can vary spatially. You simply expand around the spatially dependant $\eta$, leading to a spatially varying $\psi$. See, for example, type II superconductors Dec 4, 2017 at 10:39
• So you are saying that in the tipical arguments that leads to abelian Higg mechanism in superconductor one simply assume that $\eta$ is uniform? Dec 4, 2017 at 10:46