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We study the critical properties of the $3d$ $O(2)$ universality class inbounded domains through Monte Carlo simulations of the clock model. We use animproved version of the latter, chosen to minimize finite-size corrections atcriticality, with 8 orientations of the spins and in the presence of vacancies.The domain chosen for the simulations is the slab configuration with fixedspins at the boundaries. We obtain the universal critical magnetization profileand two-point correlations, which favorably compare with the predictions of thecritical geometry approach based on the Yamabe equation. The main result isthat the correlations, once the dimensionful contributions are factored outwith the critical magnetization profile, are shown to only depend on thedistance between the points computed using a metric found solving thefractional Yamabe equation. The quantitative comparison with the correspondingresults for the Ising model at criticality is shown and discussed. Moreover,from the magnetization profiles the critical exponent $\eta$ is extracted andfound to be in reasonable agreement with up-to-date results.

Critical $1$- and $2$-point spin correlations for the $O(2)$ model in $3d$ bounded domains