= Infrared spin-wave correlations of the XY model
{title2=$\langle\mathbf m(x)\cdot\mathbf m(0)\rangle=\bar m^2e^{-\langle(\theta(x)-\theta(0))^2\rangle/2}$}
For the <Gaussian functional integral> with dimensionless <phase stiffness> $K$ and a microscopic cutoff $a$, the <Gaussian phase averaging> identity gives the displayed <correlation function>. In two dimensions the <phase-difference variance> is $(\pi K)^{-1}\log(r/a)+O(1)$, so the <XY model> has algebraic <spin wave> correlations with exponent $1/(2\pi K)$. In one dimension the variance is $r/K$, giving exponential decay. Above two dimensions the variance has a finite large-distance limit within the low-temperature <spin wave> approximation. The <Laplacian> Green function solves $-K\nabla^2G=\delta$; integration over a sphere fixes $G(r)=r^{2-d}/[(d-2)S_dK]+C$ for $d\ne2$ and $G(r)=-(2\pi K)^{-1}\log(r/a)+C$ for $d=2$. A fixed uniform phase and ultraviolet regularization are required to define $G$ itself; phase differences eliminate the constant.
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