# Why the correlation function of 2D classical XY model is written so?

2D classical XY model $$H = -J\cos(\theta_{i}-\theta_{j})%$$ is famous for Berezinskii-Kosterlitz-Thouless phase transition. This is because of the difference of correlation function between hot and low limit. At low limit correlation function is written as $$G(\boldsymbol{r}) \sim \left(\frac{1}{r}\right)^{\frac{1}{2\pi\beta J}}$$ I can't derive it. Please teach me.

• You can look at the nook by N. Nagaosa on QFT in condensed matter physics, chapter 3. – Chuan Chen Mar 8 '19 at 3:22
• There is a detailed answer relating to the Kosterlitz-Thouless transition at physics.stackexchange.com/questions/255909/…. This particular result comes from a low-temperature spin-wave analysis, which you can also find in standard textbooks such as Chaikin and Lubensky Principles of Condensed Matter Physics. – user197851 Mar 8 '19 at 9:48