Goldstone-mode effective free energy

For a broken $O(2)$ symmetry with order-parameter magnitude $v$, the long-wavelength phase field has free energywhere ${\rho }_s$ is the stiffness, equal to $\gamma v^2$ in the simplest Landau-Ginzburg model.

For a Gaussian phase field of stiffness ${\rho }_s$,Its infrared divergence rules out conventional long-range order in dimensions at or below two.