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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 ₒ and the dimensionless wrapping probability R. From the finite-size scaling analyses of ₒ and of R, we determine the critical points as T₂ (XY) =2. 2010. 16em{0ex}8440. 16em{0ex}1 (5) and T₂ (Villain) =0. 3330. 16em{0ex}0670. 16em{0ex}04 (7) and (t/U) ₂ (BH) =0. 0590. 16em{0ex}7290. 16em{0ex}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 =0. 6710. 16em{0ex}83 (18). In addition, the critical exponent is estimated as =0. 0380. 16em{0ex}53 (48) by analyzing a susceptibilitylike 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.
Xu et al. (Wed,) studied this question.