Essential Singularity of the Correlation Length in the Planar XY Model

We prove the Berezinskii–Kosterlitz–Thouless essential singularity for the correlation length of the nearest-neighbor cosine XY model on the square lattice. Let $m(b)$ be the mass obtained by taking first the free-box thermodynamic limit and then the separation limit along a coordinate axis. As the inverse temperature b approaches b${}_c$ from the massive side, $\sqrt{b_c-b}\log \frac{1}{m(b)}⟶A_{\mathrm{XY}},$ where $A_{\mathrm{XY}}$ is a finite, strictly positive model-specific constant.