Bulk and surface properties in the critical phase of the two-dimensional XY model

Monte Carlo simulations of the two-dimensional XY model are performed in a square geometry with various boundary conditions (BC). Using conformal mappings we deduce the exponent $\eta_\sigma(T)$ of the order parameter correlation function and its surface analogue $\eta_\|(T)$ as a function of the temperature in the critical (low-temperature) phase of the model.