We calculate the von Neumann and Renyi bipartite entanglement entropy of the $O(2)$ model with a chemical potential on a 1+1 dimensional Euclidean lattice with open and periodic boundary conditions. We show that the Calabrese-Cardy conformal field theory predictions for the leading logarithmic scaling with the spatial size of these entropies are consistent with a central charge $c=1$. This scaling survives the time continuum limit and truncations of the microscopic degrees of freedom, modifications which allow us to connect the Lagrangian formulation to quantum Hamiltonians. At half-filling, the forms of the subleading corrections imposed by conformal field theory allow the determination of the central charge with an accuracy better than two percent for moderately sized lattices. We briefly discuss the possibility of estimating the central charge using quantum simulators.
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