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.
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$\begingroup$ You can look at the nook by N. Nagaosa on QFT in condensed matter physics, chapter 3. $\endgroup$– Ogawa ChenCommented Mar 8, 2019 at 3:22
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$\begingroup$ 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. $\endgroup$– user197851Commented Mar 8, 2019 at 9:48
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