Phase Transitions Of Ising Model With Cluster Weight And The Supersolid Of Bose-Hubbard Model By Monte Carlo Methods

There are many interesting physical questions and challenges in phase transitions and critical phenomena.Phase transition phenomenon can be explored by various statistical lattice models which include the Ising model,XY model,O(n)loop model and cubic loop model,etc.Those different models can also be connected together by the universal nature of the phase transition.For example,the O(n)loop model and cubic loop model have same universality on the two-dimensional square lattice in the case 0 ?n(27)2.A cluster weight Ising model is proposed by introducing an additional cluster weight in the partition function of the traditional Ising model.At the same time it is equivalent to the O(n)loop model or cubic loop model on the two dimensional lattice,but on the three dimensional lattice,it is still not very clear whether these models have the same universality.In order to simulate the cluster weight Ising model and search for new universality class,we design a highly efficient cluster algorithm,by combining the color-assignation and the Swemdsen-Wang methods.Phase transition can be divided into classical phase transition and quantumphase transition.And the phase transition of Ising model with cluster weight belongs to the classical phase transition,besides,the quantum phase transition is also studied.Bosons is one type of the micro-particles in the nature and bosons can be realized as Bose-Einstein condensates in the optical lattices.There are many quantum phases,such as superfluid and supersolid.The supersolid is a novel quantum state with diagonal long-range order(DLRO)and off-diagonal long-range order(ODLRO)simultaneously.In resent years,great progress has taken place in search of supersolid.For instance,supersolidity was found for hardcore bosons on the one-layered triangular lattice,as well as pair-supersolid on the double layer lattice,pairing tunneling supersolid,f wave supersolid and molecular supersolid on the triangular lattice.Through the stochastic series expansion directed loop quantum Monte Carlo method,we simulated the hard-core Bose-Hubbard model on the three-leg ladder.The results can be compared with those in Ref[J Phys Soc Jpn.83,064003(2014)],which studied the hard-core Bose-Hubbard model on the layered triangular lattices and found the supersolid phase based on the density wave phase along the z direction.The main content of this thesis is arranged as follows:In the first chapter,we introduce the background knowledge of phase transition,critical phenomenon,the cluster weight Ising model,the Bose-Hubbard model and show the concept of order parameter,universality,supersolid,etc.In the second chapter,we introduce the Monte Carlo method including boththe classical and the quantum methods.We explain theoretical basis,basic ideas and simulation steps.Several common algorithms in the statistical physics is discussed,such as Swemdsen-Wang algorithm,color-assignation algorithm.We also show cluster algorithm and sampled quantities of the cluster weight Ising model and the hard-core Bose-Hubbard model,respectively.In the third chapter,we design a highly efficient cluster algorithm to simulate the cluster weight Ising model on the three dimensional lattice,simulation analysis of numerical results is given.Through the simulation and calculation,we find that critical temperature Tc,the thermal exponent yt,and the magnetic exponent ym of the order parameter n=1,1.5,1.6,1.8,1.9 and 2.0,and the phase transition at n=3.The universalities of the cluster weight Ising model and the O(n)loop model on the three dimensional lattice are manifestly the same if n<2,while completely different if n=2.For larger value of n=3,we find the obvious signature of the first-order phase transition by the hysteresis loops of the energy E per site and the Binder ratio Q,which is a clear sign of the first-order phase transition.In chapter IV,through the stochastic series expansion directed loop quantum Monte Carlo method,we simulated the hard-core Bose-Hubbard model on the three-leg ladder.By measuring the structured factors and the superfluid stiffness,we find that the system has the solid with density 1/2 while the supersolid along the z direction dose not exist with nonzero quantum tunneling.In chapter V,the summary and prospect of the research work in this paper is presented.