= Solution
Assume $g>0$ for stability. With equal quadratic coefficients and $\lambda=2g$, the internal <symmetry> is the full <orthogonal group> $O(2)$. In two dimensions at positive <temperature>, with short-range interactions, the <Mermin-Wagner theorem> forbids <spontaneous symmetry breaking> of this <continuous symmetry>. Thus $\langle\phi_1\rangle=\langle\phi_2\rangle=0$ in both phases in the infinite system; a nonzero average <order parameter> does not distinguish them.
Instead measure the <correlation function>
$$
G(r)=\langle\boldsymbol\phi(x)\cdot\boldsymbol\phi(0)\rangle,\qquad r=|x|.
$$
\b[The low-temperature phase has <quasi-long-range order> and algebraic correlations; the high-temperature phase has a finite <correlation length> and exponential correlations.] More precisely,
$$
\boxed{G(r)\sim r^{-\eta(T)}\ \text{below }T_{\mathrm{BKT}},\qquad G(r)\ \text{decays exponentially for }T>T_{\mathrm{BKT}}}.
$$
A finite <order parameter> amplitude allows $\boldsymbol\phi\simeq v(\cos\theta,\sin\theta)$. Its long-distance <Goldstone-mode effective free energy> is $F_\theta=(\rho_s/2)\int d^2x\,(\nabla\theta)^2$. Restoring the <Boltzmann constant> and using the renormalized <phase stiffness>, the <thermal phase fluctuations> give
$$
\langle[\theta(x)-\theta(0)]^2\rangle\simeq\frac{k_BT}{\pi\rho_s}\log(r/a),\qquad \eta(T)=\frac{k_BT}{2\pi\rho_s}.
$$
The <Berezinskii–Kosterlitz–Thouless transition> separates <vortex-antivortex binding> from unbound <vortex> defects. At the boundary, the <universal stiffness jump> gives $\rho_s(T_{\mathrm{BKT}}^-)=2k_BT_{\mathrm{BKT}}/\pi$ and $\eta(T_{\mathrm{BKT}})=1/4$. The low-temperature <phase stiffness> is nonzero; it vanishes at long distances in the disordered phase. The exact transition location is fluctuation dependent and cannot be obtained just by setting the bare $\mu^2$ to zero.
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