= Solution
The dot product of the two <magnetizations> is $\bar m^2\cos[\theta(x)-\theta(0)]$. For the Gaussian <spin wave> field, the <Gaussian phase averaging> identity $\langle e^{iX}\rangle=e^{-\langle X^2\rangle/2}$ for a centered Gaussian variable gives
$$
\langle\mathbf m(x)\cdot\mathbf m(0)\rangle
=\bar m^2\exp[-D(r)/2],\qquad
D(r)=\langle[\theta(x)-\theta(0)]^2\rangle=2[G(0)-G(r)].
$$
Keeping this exponential is essential: expanding the cosine to quadratic order would fail when phase differences grow with separation. Equivalently, the <phase-difference variance> is
$$
D(r)=\frac2{\bar K}\int\frac{d^dq}{(2\pi)^d}\frac{1-\cos(q\cdot x)}{q^2}.
$$
For $d>2$, the integral defining the absolute phase variance is finite at small momentum after fixing the uniform mode; its microscopic <ultraviolet cutoff> still matters. Its nonconstant <Green function> decays as $r^{2-d}$, so $D(r)$ tends to a finite constant $D_\infty$ and
$$
\boxed{\lim_{r\to\infty}\langle\mathbf m(x)\cdot\mathbf m(0)\rangle
=m_0^2=\bar m^2e^{-D_\infty/2}>0\quad(d>2).}
$$
In two dimensions, the logarithmic <Green function> gives $D(r)=(\pi\bar K)^{-1}\log(r/a)+O(1)$. Consequently
$$
\langle\mathbf m(x)\cdot\mathbf m(0)\rangle\asymp\bar m^2(r/a)^{-1/(2\pi\bar K)}\longrightarrow0.
$$
For $1\le d<2$, $D(r)$ grows as $2r^{2-d}/[(2-d)S_d\bar K]$, up to bounded microscopic terms, and the <correlation function> again vanishes. In particular, in one dimension $S_1=2$ gives $D(r)=r/\bar K$ and exponential decay $\bar m^2e^{-r/(2\bar K)}$. Therefore \b[there is no conventional long-range order for $d\le2$ at positive temperature in this short-range continuous-symmetry model]. In two dimensions, algebraic decay can still support <quasi-long-range order>.
These <infrared spin-wave correlations of the XY model> explain the mechanism behind the <Mermin-Wagner theorem>. They are a test of the low-temperature <spin wave> approximation, not a claim that every $d>2$ <XY model> is ordered at every temperature. Large local fluctuations and defects destroy the ordered phase at sufficiently high temperature. Also, the assumed small angle is a local smooth-field approximation; phase differences at arbitrarily large separation need not remain small.
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