Monte Carlo Simulations Of Critical Behavior In Lattice Models:Emergent Symmetries,Universal Dynamics,and Upper Critical Dimensionality

Lattice models are typical platforms for the study of phase transition and have a close relationship with several real physical systems.In this thesis,we study typical critical phenomena of lattice models utilizing high-efficiency Monte Carlo simulations,paying particular attention to the critical behaviors that are relevant to emergent symmetry,universal dynamics,and upper critical dimensionality.In the Introduction,we firstly introduce the concepts of the phase transition and critical behavior,the Monte Carlo simulations,and the finite-size scaling.We then introduce lattice models that are relevant to our studies,which include the clock model,Villain model,Bose-Hubbard model,dimer model,and percolation model.We also review the Monte Carlo methods adopted in this thesis.This kind of method is an efficient numerical methodology based on random sampling and known as a state-of-the-art method for the study of phase transitions and critical phenomena in many-body systems.In the subsequent sections,we describe the studies which include:(1)By using the flow representation and the worm-type Monte Carlo method,we study the critical behavior of the six-state clock model on simple cubic lattices.We measure several wrapping probabilities via the topological structure of flow configurations on the torus.The relation between wrapping probabilities and temperature is analyzed.On this basis,we perform scaling analyses on the finite-size Monte Carlo data by which we obtain the precise results of the critical point and the magnetic critical exponent.It is found that the estimated critical exponent is consistent with that in the three-dimensional U(1)universality class.Hence,we confirm in the flow representation that the three-dimensional six-state clock model has an emergent U(1)symmetry.(2)We analyze the linear response dynamics of typical lattice models in the(2+1)dimensional U(1)universality class.The Villain model and the Bose-Hubbard model are considered.On one hand,we use the classical worm algorithm in flow representation to simulate the simple-cubic Villain model.On the other hand,we perform simulations for the square-lattice Bose-Hubbard model at unit filling,using the worm-type quantum Monte Carlo algorithm in the path-integral representation.We obtain precise Monte Carlo data for large systems at low temperatures and perform scaling analyses.We observe the universal dynamics of(2+1)-dimensional U(1)critical systems,confirming the unitary scaling relation among the imaginary-frequency conductivity,the frequency number and the temperature.(3)In order to explore the emergent O(5)symmetry,we analyze the extended dimer model on simple-cubic lattices.We employ the worm algorithm to simulate the model in an extensive parameter regime and confirm the estimates of phase transition points.On this basis,we perform scaling analyses to the finite-size Monte Carlo data and obtain the estimates of renormalization critical exponent y_t.Hence,we confirm the existence of emergent symmetry.Besides,we observe a crossover behavior between the non-extended regime and the extended regime.(4)In order to explore the logarithmic finite-size scaling at the upper critical dimensionality,we measure the wrapping probabilities,the susceptibility-like quantities,and the averaged sizes of clusters in the six-dimensional percolation model.We analyze the finite-size Monte Carlo data and confirm the existence of logarithmic corrections.Besides,we find that the physical quantities without incorporating the largest cluster exhibit minor high-order corrections.A renormalization group prediction of logarithmic exponent is therefore confirmed.