  Motivated by a recent experiment on a square-lattice Rydberg atom array
realizing a long-range dipolar XY model [Chen et al., Nature (2023)], we
numerically study the model's equilibrium properties. We obtain the phase
diagram, critical properties, entropies, variance of the magnetization, and
site-resolved correlation functions. We consider both ferromagnetic and
antiferromagnetic interactions and apply quantum Monte Carlo and
pseudo-Majorana functional renormalization group techniques, generalizing the
latter to a U(1) symmetric setting. Our simulations perform extensive
thermometry for the first time in dipolar Rydberg atom arrays and establish
conditions for adiabaticity and thermodynamic equilibrium. On the ferromagnetic
side of the experiment, we determine the entropy per particle S/N~0.5, close to
the one at the critical temperature, S_c/N = 0.585(15). The simulations suggest
the presence of an out-of-equilibrium plateau at large distances in the
correlation function, thus motivating future studies on the non-equilibrium
dynamics of the system.
