= Solution
The <magnetization> correlator is an average of a <phase difference>, not merely its lowest Taylor expansion. <Gaussian phase averaging> gives
$$
C_m(r)=\bar m^2\langle\cos[\theta(\mathbf x)-\theta(0)]\rangle
=\bar m^2\exp\!\left[-\frac12D(r)\right],
$$
where
$$
D(r)=\langle[\theta(\mathbf x)-\theta(0)]^2\rangle
=\frac2{\bar K}\int\frac{d^dq}{(2\pi)^d}\frac{1-\cos(\mathbf q\cdot\mathbf x)}{q^2}
=2[G(0)-G(r)].
$$
The coincident value is ultraviolet-regularized at a microscopic scale $a$. At large $r$:
* For $d>2$, the infrared integral for $G(0)$ converges. Choosing $G(\infty)=0$, $D(\infty)=2G(0)<\infty$, so \b[$C_m(\infty)=m_0^2=\bar m^2e^{-G(0)}>0$] within the low-temperature <spin wave> <thermodynamic phase>.
* For $d=2$, $D(r)=(\pi\bar K)^{-1}\log(r/a)+O(1)$, hence \b[$C_m(r)\propto r^{-1/(2\pi\bar K)}\to0$].
* For $0<d<2$, the leading growth is $D(r)=2r^{2-d}/[(2-d)S_d\bar K]+O(1)$, hence \b[$C_m(r)\propto\exp\{-r^{2-d}/[(2-d)S_d\bar K]\}\to0$]. In particular $d=1$, $S_1=2$, gives $C_m(r)\propto e^{-r/(2\bar K)}$.
\Image[/past-exam-of-the-phase-transitions-and-collective-phenomena-course-of-the-university-of-cambridge/2012/paper-1-ptcp2012-correlations.png]
{title=Infrared spin-wave correlation limits}
{description=Original regulated phase-only model curves for dimensions one, two and three. They illustrate exponential decay, algebraic decay and a nonzero limit respectively; no experimental dataset or phase-transition temperature is fitted.}
Thus \b[two is the <lower critical dimension> for conventional positive-temperature continuous-symmetry order] under the short-range assumptions. Two dimensions can still have <quasi-long-range order> and a thermodynamic transition; zero limiting <magnetization> does not forbid them. The $d>2$ <spin wave> result describes an admissible low-temperature ordered <thermodynamic phase>, not order at arbitrarily high <temperature>. The <Mermin-Wagner theorem> supplies the more general no-order statement below or at two dimensions.<Statistical Field Theory by David Tong>[§4.2.3: infrared phase fluctuations and continuous-symmetry order]{ref}
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