The Study Of Thermodynamic Limit And Boundary Energy Of The Spin-1 Isotropic Heisenberg Spin Chain

The quantum integrable system is a kind of significant physical model and the related research was applied in various fields of physics.The study of the exact solution of these models can provide a good explanation of some physical phenomena and concepts,and also give the benchmark for many important physical experiments.Therefore,it is of great importance to investigate the quantum integrable model.The Heisenberg spin chain model is one of the usual quantum integrable models,and the study of high spin chain models is applied in many fields such as quantum field theory and condensed matter physics.In this thesis,the model we focused on is the 1-D spin-1 isotropic Heisenberg spin chain,based on the exact solution of the model with periodic boundary and off-diagonal boundary conditions,the ground state energy and boundary energy of the model in the thermodynamic limit is obtained combined with numerical.Firstly,we give the introduction of the quantum integrable system and introduce the methods to solve the quantum integrable models exactly by using some examples.The algebraic Bethe Ansatz method for solving the models with U(1)symmetry exactly was introduced by using the spin-1/2 chain model with periodic boundary conditions.As for the models without U(1)symmetry,the off-diagonal Bethe Ansatz method was introduced by using the spin-1/2chain with arbitrary boundary conditions.Then we calculated the ground state energy and boundary energy of the spin-1 chain in the thermodynamic limit by numerical based on the exact solution of it obtained by the methods introduced above.For the spin-1 chain with periodic boundary condition which satisfies U(1)symmetry,the exact solution of the model can be solved by using the algebraic Bethe Ansatz method,and the ground state energy of the system in the thermodynamic limit can be calculated by using the Yang-Yang thermodynamic method.As for the spin-1 chain with off-diagonal boundary condition in which U(1)symmetry is broken,the conventional methods cannot be applied anymore,the model can be solved exactly by using the off-diagonal Bethe Ansatz method.Due to the effect of the unparallel boundary,the model corresponds to the inhomogeneous T-Q relation and the Yang-Yang thermodynamic method cannot be applied directly.By using numerical analysis,we find that the effect of the inhomogeneous term to the system can be neglected in the thermodynamic limit,the inhomogeneous T-Q relation is reduced to the homogeneous T-Q relation in the thermodynamic limit.Thus,we calculated the ground state energy and boundary energy of the system by using the Yang-Yang thermodynamic method,these results coincide with DMRG numerical results very well and show the correctness of the conclusion.These results can be generalized to the SU(2)symmetric high spin Heisenberg model directly.