Phase-difference variance

For the physical free energythe Boltzmann factor is $e^{-F_{\theta }/(k_{B}T)}$. At $T>0$, the Gaussian thermal phase fluctuations therefore obeyThe temperature factor is necessary when ${\rho }_s$ is the energy stiffness defined by the parent. Defining a dimensionless stiffness $K={\rho }_s/(k_{B}T)$ instead replaces $k_{B}T/{\rho }_s$ by $1/K$.

The difference cancels the spatially uniform phase mode. At any fixed separation, $1-\cos (\mathbf{k}\cdot \mathbf{x})$ removes the small-$k$ singularity. What grows in low dimensions is the variance as the separation tends to infinity.

In one dimension, the integral equals $|x|/2$, so the variance is $(k_{B}T/{\rho }_s)|x|$. In two dimensions, with microscopic cutoff $a$ and $r\gg a$, it grows asFor the Gaussian field, the order parameter correlation isIt decays exponentially in one dimension and algebraically in two. Thus nonzero-temperature Gaussian fluctuations do not leave conventional long-range order in these dimensions. This is the long-wavelength mechanism related to the Mermin-Wagner theorem; vortices and interactions require the fuller theory of the Berezinskii–Kosterlitz–Thouless transition in the two-dimensional case.

Fix the uniform order-parameter phase and retain finite system size $L$ before evaluating a massless Gaussian functional integral. In $d\le 2$, its coincident and separated order-parameter phase covariances each acquire a divergent additive constant as $L\to \infty$. Their difference $2[G_L(0)-G_L(r)]$ instead has a finite fixed-separation limit for $d>0$. This is why physical XY model correlations are obtained from Gaussian phase averaging of angle differences. An ultraviolet cutoff also remains necessary for coincident-point quantities; infrared cancellation does not remove it. The logarithmic two-dimensional Green function follows from the $d\to 2$ power-law expression only after subtracting its constant pole.

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 $-K{\nabla }^{2}G=\delta$; integration over a sphere fixes $G(r)=r^{2-d}/[(d-2)S_{d}K]+C$ for $d\ne 2$ 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.