The critical logarithmic correction for the planar XY model

We prove the critical logarithmic correction for the nearest-neighbor cosine XY model on the square lattice. Let $C_{b_c}(r)$ be the two-point correlation at critical inverse temperature, obtained by taking the free-box thermodynamic limit while the two sites remain r lattice steps apart along a coordinate axis. Then $C_{b_c}(r)=B_{\mathrm{XY}}{r}^{-1/4}(\log r{)}^{1/8}(1+o(1)),B_{\mathrm{XY}}\in (0,\infty ),$ as $r\to \infty$.