We present a worm-type Monte Carlo study of several typical models in the three-dimensional (3D) U(1) universality class, which include the classical 3D XY model in the directed flow representation and its Villain version, as well as the 2D quantum Bose-Hubbard (BH) model with unitary filling in the imaginary-time world-line representation. From the topology of the configurations on a torus, we sample the superfluid stiffness $rho_s$ and the dimensionless wrapping probability $R$. From the finite-size scaling analyses of $rho_s$ and of $R$, we determine the critical points as $T_c ({rm XY}) =2.201, 844 ,1(5)$ and $T_c ({rm Villain})=0.333, 067, 04(7)$ and $(t/U)_c ({rm BH})=0.059 , 729 ,1(8)$, where $T$ is the temperature for the classical models, and $t$ and $U$ are respectively the hopping and on-site interaction strength for the BH model. The precision of our estimates improves significantly over that of the existing results. Moreover, it is observed that at criticality, the derivative of a wrapping probability with respect to $T$ suffers from negligible leading corrections and enables a precise determination of the correlation length critical exponent as $ u=0.671 , 83(18)$. In addition, the critical exponent $eta$ is estimated as $eta=0.038 , 53(48)$ by analyzing a susceptibility-like quantity. We believe that these numerical results would provide a solid reference in the study of classical and quantum phase transitions in the 3D U(1) universality, including the recent development of the conformal bootstrap method.
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