= Solution
When $m_1=m_2=\mu^2$, the <free energy> has <continuous symmetry> $O(2)$. Its ordered minima satisfy
$$
\rho_0^2=\phi_1^2+\phi_2^2=-\frac{\mu^2}{4g}.
$$
Continuous phase fluctuations make the lower critical dimension $d_{\rm l}=2$, in agreement with the <Mermin-Wagner theorem>. In two dimensions write the complex <order parameter> as $\psi=\rho_0e^{i\theta}$. Neglecting the massive <amplitude mode> gives the <Goldstone-mode effective free energy>
$$
F_\theta=\frac{\rho_0^2}{2}\int d^2x\,(\nabla\theta)^2.
$$
The <phase-difference variance> is
$$
\langle[\theta(x)-\theta(0)]^2\rangle
=\frac1{\pi\rho_0^2}\log\frac r a+O(1),
$$
and therefore
$$
\langle\psi(x)\psi(0)^*\rangle
\sim \rho_0^2\exp\left[-\frac12
\langle(\theta(x)-\theta(0))^2\rangle\right]
\sim r^{-\eta},
$$
with
$$
\boxed{\eta=\frac1{2\pi\rho_0^2}=-\frac{2g}{\pi\mu^2}.}
$$
Thus <spin waves> replace true long-range order by <quasi-long-range order>.
A <vortex>-antivortex pair of separation $R$ has the logarithmic energy $E_{\rm pair}\simeq2\pi\rho_0^2\log(R/a)$. The number of pair separations below $R$ grows as $R^2$, giving the coarse entropy $S_{\rm pair}\simeq2\log(R/a)$. Thus
$$
F_{\rm pair}\simeq2(\pi\rho_0^2-1)\log(R/a),
$$
and widely separated pairs become favorable at $\rho_0^2\simeq1/\pi$. Substituting the mean-field stiffness gives
$$
\boxed{\mu^2\simeq-\frac{4g}{\pi},}
$$
suggesting a <Berezinskii–Kosterlitz–Thouless transition>. The numerical location is only a bare-stiffness estimate: vortex-core fluctuations renormalize the stiffness near the transition.
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