Monte Carlo Simulations Of (2+1)-and(3+1)-dimensional U(1) Critical Systems

The U(1)criticality is a paradigmatic testbed for the study of phase transition and plays a crucial role in various many-body phenomena.These many-body phenomena include not only vortex binding-unbinding transition,superfluid and spin liquid,but also the relativistic gauge fields in particle physics and the emergent continuous symmetries responsible for deconfined criticality.Nevertheless,there are some fundamental questions about the U(1)criticality which need clarifications.In(2+1)-dimensions,the estimates of critical exponents by space experiments and by numerical simulations are not compatible.In(3+1)-dimensions,the U(1)systems are at the upper critical dimensionality where the mean-field theory should be modified by logarithmic corrections.The precise picture for the coexistence of logarithmic corrections and mean-field mechanism remains an open question.In this thesis,we adopt efficient Monte Carlo methods to study the U(1)criticality of paradigmatic lattice systems.In the introduction,we firstly describe two lattice models,namely,the O(n)model and the Bose-Hubbard model.A special case for the former is equivalent to the U(1)model,and the latter is a paradigmatic host for the U(1)quantum criticality.We briefly describe phase transitions and critical phenomena,together with their finite-size scaling.Moreover,we introduce the Monte Carlo methods adopted throughout the thesis.In what follows,we shall report our researches through the following aspects:(1)We perform worm-type Monte Carlo simulations for several typical models in the(2+1)-dimensional U(1)universality class,which include the classical three-dimensional XY model in the directed flow representation and its Villain version,as well as the two-dimensional quantum Bose-Hubbard model at unit filling in the imaginary-time world-line representation.From the topology of the configurations on a torus,we sample the superfluid stiffness ?s and the dimensionless wrapping probability R.Using the finite-size scaling analyses of ?s and of R,we obtain precise estimates of critical points that improve significantly over that of the existing results.Meanwhile,our estimates of critical exponents have a comparable precision with literature results.We believe that these independent numerical results provide a solid reference in the study of classical and quantum phase transitions in the(2+1)-dimensional U(1)universality,and for the recent development of the conformal bootstrap method(2)Aiming at exploring the finite-size scaling for the(3+1)-dimensional U(1)critical systems,we study the O(n)vector models on periodic four-dimensional hypercubic lattices.We find that the two-point correlation g(r,L),with L the linear size,exhibits a novel two-scale behavior:following a mean-field behavior r-2 at shorter distance and entering a plateau at larger distance whose height decays as L-2(lnL)p with p a logarithmic correction exponent.This finding is further supported by the finite-size scaling of the magnetic susceptibility,the magnetic fluctuations at non-zero Fourier modes,and the two-point correlation g(L/2,L).Our observation suggests that in consistent with recent experiments and quantum Monte Carlo simulations,the theoretically predicted logarithmic reduction is probably absent in the width-mass ratio of Higgs resonance as arising from the longitudinal fluctuations around the(3+1)-dimensional O(n)quantum critical point.(3)We study the universal finite-size and finite-frequency scaling for the dynamics of the(3+1)-dimensional U(1)critical systems.Firstly,we perform worm-type Monte Carlo simulations for the Villain model on four-dimensional hypercubic lattices and the Bose-Hubbard on simple-cubic lattices,sampling superfluid stiffness and winding probability.For the finite-size Monte Carlo data of the superfluid stiffness and the winding probability,we present a two-component finite-size scaling that produces an accurate estimate of critical points.On this basis,we further compute the critical conductivity ?(i?n)as a function of imaginary frequency(n is frequency index),providing strong evidence for the theoretically predicated scaling ?(i?n)=p(T)/n?-1,witl ? the spatial scaling dimension and p(T)a function of temperature.