Monte Carlo Simulations Of Phases And Phase Transitions In Lattice Boson (Spin) Systems

Phase transitions,widely exist in nature,reflects the competitions between interactions and thermal or quantum fluctuations.As a result,physical quantities generally become discontinuous or divergent at a transition point.Lattice boson?spin?models are characteristic many-body systems and important hosts of phase transitions and critical phenomena.Lattice boson systems?e.g.,Bose-Hubbard model?and their typical phase transitions have been simulated in optical lattices with cold atoms.As an unbiased and efficient numerical method,Monte Carlo method is widely adopted in the theoretical study of lattice boson?spin?models.In recent years,the combination of Monte Carlo simulations and machine learning,machine learning Monte Carlo?MLMC?,provides a new view for studying phase transitions and critical phenomena.In this thesis,phases and phase transitions of lattice boson?spin?models are explored by Monte Carlo simulations.Firstly,an introduction of two important many-body systems---Ising model and Bose-Hubbard model---are presented?our work is based on the extensions of these two models?.An introduction of critical phenomena and Monte Carlo simulations also are included.We then present our numerical results for the phases and phase transitions of certain lattice boson?spin?models,which are outlined as follows:?1?We construct a model of interacting hard-core bosons on a squarelatticewithhoppingsofdifferentlengths,featuring nearest-neighbor?NN?hopping?t₁?,anisotropic next nearest-neighbor?NNN?hopping?t?₂?,and nearest-neighbor repulsion?₁V?.This model is calledt₁-t?₂-V₁model.On the basis of large-scale quantum Monte Carlo simulations,we establish a ground-state phase diagram and a finite-temperature phase diagram for the model.We present numerical evidences for the existence of checkerboard supersolid state?CSS?and explore the stability of CSS phase against thermal fluctuations.We observe a counterintuitive order-by-disorder behavior and provide an entropy criterion to explain the behavior.?2?We use a machine learning Monte Carlo method based on the principal component analysis?MLMC-PCA?to study the phase transitions and critical phenomena of the Ashkin-Teller?AT?model.Compared with conventional Monte Carlo methods,MLMC-PCA can correctly recognize the phase transitions of AT model even without prior knowledge about order parameters.We find the correspondence between the results of MLMC-PCA and conventional Monte Carlo methods.These results indicate that the MLMC-PCA provides a new perspective for the numerical study of phase transitions and serves as a complement to existing methods.?3?To explore the possibility that the MLMC-PCA recognizes the non-local string order in the bosonic systems,we study the phase transitions from charge density wave?CDW?to Haldane insulator?HI?to superfluid?SF?in the one-dimensional extended Bose-Hubbard model.Being fed with particle number configurations,MLMC-PCA can not completely recognize these phase transition behaviors.We then make a non-local-to-local mapping and succeed in a complete recognition among the CDW,HI and SF phases.