The Critical Spin Field of the Planar XY Model

Using the companion critical Bessel-height and pin-limit theorems, we prove that at the mass-defined critical inverse temperature of the nearest-neighbor XY model on the square lattice, the spin field on a square with boundary angles fixed to zero converges along the full sequence to the full-variance imaginary exponential of a zero-Dirichlet Gaussian free field with stiffness $2/\pi$. The normalization uses the exact center magnetization and the lattice Green-function factor. Convergence holds in law in $H_{\mathrm{loc}}^{-3}((-1,1{)}^2)$; all mixed moments and joint laws of fields smeared against smooth compactly supported test functions in $(-1,1{)}^2$ converge as well.