 Solution

When $m_1=m_2={\mu }^2$, the free energy has continuous symmetry $O(2)$. Its ordered minima satisfyContinuous phase fluctuations make the lower critical dimension $d_{\mathrm{l}}=2$, in agreement with the Mermin-Wagner theorem. In two dimensions write the complex order parameter as $\psi ={\rho }_0{e}^{i\theta }$. Neglecting the massive amplitude mode gives the Goldstone-mode effective free energyThe phase-difference variance isand thereforewithThus spin waves replace true long-range order by quasi-long-range order.

A vortex-antivortex pair of separation $R$ has the logarithmic energy $E_{\mathrm{pair}}\simeq 2\pi {\rho }_0^2\log (R/a)$. The number of pair separations below $R$ grows as $R^2$, giving the coarse entropy $S_{\mathrm{pair}}\simeq 2\log (R/a)$. Thusand widely separated pairs become favorable at ${\rho }_0^2\simeq 1/\pi$. Substituting the mean-field stiffness givessuggesting a Berezinskii–Kosterlitz–Thouless transition. The numerical location is only a bare-stiffness estimate: vortex-core fluctuations renormalize the stiffness near the transition.