 Solution

For an infrared-regulated phase covariance, fix the uniform order-parameter phase zero mode and introduce ultraviolet and infrared cutoffs before taking limits. With the Fourier conventionthe dimensionless quadratic Hamiltonian diagonalizes asCompleting the square with a source gives $Z[J]/Z[0]=\exp [\frac{1}{2}\int JGJ]$, with inverse kernelThere is no additional $k_{B}T$ factor: it is already included in $\bar{K}$ because the supplied Hamiltonian is $\beta H$.

It is simplest to evaluate the real-space dependence by its defining Green equation, $-\bar{K}{\nabla }^{2}G={\delta }^{(d)}$. Rotational invariance gives $r^{d-1}{G}^{\prime }(r)=$ constant for $r>0$. Integrating over a small sphere fixes $-\bar{K}{S}_d{r}^{d-1}{G}^{\prime }(r)=1$, where $S_d=2{\pi }^{d/2}/\Gamma (d/2)$. Integrating once more,In two dimensions, take the limit after absorbing the divergent $r$-independent piece into $C$:For $d\le 2$, $G$ itself has an infrared-divergent constant in the infinite system; it is not a finite thermodynamic covariance without a regulator. The physical phase-difference variance is finite at fixed separation and cancels that constant. These are smooth spin wave correlations; compact-field vortex sectors are considered separately below.