Goldstone-mode effective free energy
= Goldstone-mode effective free energy
{c}
For a broken $O(2)$ symmetry with order-parameter magnitude $v$, the long-wavelength phase field has free energy
$$
F_\theta=\frac{\rho_s}{2}\int d^dx\,(\nabla\theta)^2,
$$
where $\rho_s$ is the stiffness, equal to $\gamma v^2$ in the simplest Landau-Ginzburg model.