$XY$-model Green function

Question

Considering the two-dimensional XY-model, how should I compute the Green function $G(\vec{r})=\left\langle\phi(0)\phi(r)\right\rangle$ -the scalar field $\phi$ denotes the orientation of the planar spins with respect to some reference axis- from the simple Poisson equation

$$-\beta J\nabla^{2}G(\vec{r})=\delta(\vec{r}),$$

having defined the rotational-symmetric "electric field" $\vec{E}(\vec{r})=\frac{\rho}{2\pi\mid{\vec{r}\mid^{2}}}\vec{r}$ analagously to electrodynamics, where $\nabla_{\vec{r}} G(\vec{r})=\vec{E}(\vec{r})$? I would like to obtain the desired logarithmic behaviour, which is then inserted in the exponential in order to obtain the correlation function and its algebraic decay, without having to deal with the nasty Bessel function integral used in each reference I found by now.

Answer 1

The difficulty in the calculation is that $\phi$ is an angular variable, so that $\phi \equiv \phi+2\pi$. There's no way to avoid the nasty calculations if you want the exact result.

However, if you're willing to pretend that $\phi$ ranges on $(-\infty, +\infty)$, then the equation you wrote down is the same as the equation for the 2D electrostatic potential ($G$, in the analogy) due to a point charge. Equivalently, it is the potential for a line charge in 3D. The calculation of this is standard, and can be found in many textbooks.

The good news is that the result you get this way is the leading order result for the case where $\phi$ is angular, (as explained for example in John Cardy's Scaling and Renormalisation in Statistical Physics) and it's good enough in many cases.