# Correlation function of XY-model with vortices

The action for XY-model without magnetic field (in continuum limit) is $$S=\int d^2x\,(\partial_{\mu}\phi)^2,$$ which gives the following motion equation: $$\Delta\phi=0.$$ Single-valued, smooth solutions of this equations are called spin-waves. Then, multi-valued field configurations (i.e. vortices) are denoted by $\Theta$. One can plug vortices solutions into $S$ and calculate it (boundary term is zero). This action one can denote by $S_{\text{Cb}}$. Taking all derivations into account, the correlation function can be written as $$\left\langle e^{i(\phi(x)+\Theta(x))}e^{-i(\phi(y)+\Theta(y))}\right\rangle_{\mathcal{Z}_{\text{sw}}\times\mathcal{Z}_{\text{Cb}}}=\left\langle e^{i(\phi(x)-\phi(y))}\right\rangle_{\mathcal{Z}_{\text{sw}}}\left\langle e^{i(\Theta(x)-\Theta(y))}\right\rangle_{\mathcal{Z}_{\text{Cb}}}.$$

My questions are:

1. Can be the $\langle...\rangle_{\mathcal{Z}_{\text{Cb}}}$ calculated exactly?
2. Can I see the BKT-transition from the two-point correlation function?