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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You can look at the nook by N. Nagaosa on QFT in condensed matter physics, chapter 3. – Ogawa Chen Mar 08 '19 at 03:22
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There is a detailed answer relating to the Kosterlitz-Thouless transition at https://physics.stackexchange.com/questions/255909/what-is-the-kosterlitz-thouless-transition. 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. – Mar 08 '19 at 09:48