Tensor Network Simulation For The J₁-J₂ Clock-3 Model On The Square Lattice

The strict Onsager solution of the 2-D Ising model provides rich and profound content for the phase transition.Understanding complex phenomena by simulating simple classical statistical models is at the heart of physics.Such practical work has achieved great success in the study of continuous phase transitions.Phase transitions are also common in life,such as the formation of ice.In order to understand phase and phase transitions,we want to simulate these systems and get numerical solutions.However,because the number of degrees of freedom of a many body system increases exponentially with the increase of the system,these solutions cannot be obtained in most cases.Many numerical and analytical methods have been developed to solve this problem.The more common numerical methods include exact diagonalization(ED),density matrix renormalization group(DMRG),and Monte Carlo(MC).In recent decades,tensor networks have gained great development as a tool in the field of many body physics.For example,tensor renormalization group algorithm(TRG),second tensor renormalization group algorithm(SRG),higher order tensor renormalization group(HOTRG)and higher order second tensor renormalization group(HOSRG)based on coarse-grained summation method,etc.There are also transfer matrix renormalization group(TMRG)based on the transfer matrix contraction method,the time evolution block decimation algorithm(TEBD)and corner transfer matrix renormalization group algorithm(CTMRG)used in this paper,etc.In recent years,people have shown great enthusiasm in the research of q-state clock model,because it shows rich physical content.As q??,the clock model is equivalent to the XY model,so it is considered to be a discrete XY model.In this thesis,we use the corner transition matrix renormalization group(CTMRG)based on the tensor network algorithm to study the three-state clock model with nearest neighbors interaction₁and next nearest neighbors interactiona₂,and explore the phase transition in the system and the mechanism behind the phase transition.The first chapter of this paper introduces the basic knowledge and representation of tensor network,and introduces the CTMRG algorithm based on tensor network representation in detail.Finally,we introduce how to calculate some thermodynamic values.The second chapter mainly introduces q-state clock model.For the classical clock statistical model,some characteristics of the model are explained firstly,and the phase transformation of q state clock model is introduced.And briefly introduces two other models closely related to q-state clock model,namely classical statistical Potts model and XY model.The third chapter is the core content of this paper.Firstly,we briefly introduce theJ₁-J₂ q-state clock model which we want to study.Then,we will analyze the calculation results of CTMRG method.Through the analysis of the dependence graph between the thermodynamic values of energy,free energy,entropy,magnetization and specific heat dependent on temperature,the type of phase transition and the mechanism behind the phase transition are determined.The fourth part of this paper is the summary of this paper,and the prospect of using other numerical methods based on tensor network to study the q-state clock model.