High-precision Monte Carlo study of several models in the three-dimensional U(1) universality class

TL;DR

This study uses high-precision Monte Carlo simulations to accurately determine critical points and exponents in the 3D U(1) universality class across classical and quantum models, improving existing estimates.

Contribution

It provides highly precise critical parameters and exponents for several models in the 3D U(1) class, surpassing previous results and aiding future theoretical and numerical studies.

Findings

Critical temperatures for XY, Villain, and Bose-Hubbard models determined with high precision.

Correlation length exponent ν estimated as 0.67183(18).

Anomalous dimension η estimated as 0.03853(48).

Abstract

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 $\nu=0.671 \, 83(18)$. 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.