The Exact Solution Of Integrable System Of Long-range Correlation

In this thesis, we stud the integrable models with long range interaction, such as Gaudin model. Calogero-Sutherland model and Sutherland-Rbmer model.The first chapter include a brief introduction of the exactly solvable models of strongly correlated electrons.In the second and third chapters, we propose the rational and the trigonometric SU(N) Gaudin models. By taking the quasi-classical limit of both the quantum determinant and the transfer matrices of the anisotropic SU(N) and SL7~,(N) spin chains, we obtain the Hamiltonians and the generating functions of the Gaudiri models. We also express the eigenstates of the Gaudin models explicitly. Then using the Quantum Inverse Scattering Method (QISM), we arrive at the eigenvalues of the Hamiltonians and the generating functions of these systems. The SU(3) Gaudin model are constructed based on the quasi-classical limit of the SU(3) chain under the periodic boundary conditions. Based onthe well-defined quantum determinant with SU(3) symmetry, the eigenvectors and eigenvalues of the Hamiltonians and the generating functions of the SU(3) Gaudin mode] are given. We also extend our results to the high spin case and obtain the rational and trigonometric SU(N) symmetric Gaudin models safely.Th the chapter 4, we propose a supersymmetric SU(112) Gaudin model. we present the Hamiltonians and the generating functions explicitly. By using the graded Quantum Inverse Scattering Method, we derive out the eigenvalues of these sstems. This is first to construct the graded quantum determinant with the SU(112) supersvmmetrv. We also present the well-defined eigenstates through the quasi-classical limit of the eigenstates in the supersymmetric t-J model.In the fifth chapter, we propose the Hamiltonians of the generalized SUq(1j2) Gaudin model. We choose the periodic generalized t-J model as the perfect starting point. With the help of the well-defined graded quantum determinant, we obtain the eigenstates and eigenvalues of the generating function and the Hamiltonians of the Gaudin model in fermionic background in the framework of the graded QISM. The Bethe ansatz equ~tions are also obtained.In the sixth chapter, by constructing a new Dunk] operators with intrinsic symmetry, a two-component Calogero-Sutherland model with molecular field is established. The conserved quantities are proposed and the integrability of the system is also proved. By using the asymptotic Bethe ansatz method, the exact solutions of the model is driven out. For the r2 limit of the sinh2 (r)-interaction, the rational two-body scattering matrix is recovered. Then, the thermodynamics of the model with the periodic boundary condition under the rational limit is discussed.-7In the seventh chapter, by constructing the reflection spin-Dunkl operator, the Sutherland-R鰉er model with open boundary condition is established. It describes a one-dimensional, two-component, quantum many-body system in which like particles interact with a pair potential g(g + 1)/sinh2(r), while unlike particles interact with a pair potential 梘(g + 1)/cosh2(r). We first give a proof of integrability of the system. By solving the Schr鰀inger equation and using the properties of the hypergeometric function and gamma function, we arrive at the twobody scattering matrix and the reflection matrix. Then with the help of the knowledge of inhomogeneous six-vertex model, we obtain the eigenstates, eigenvaiues and the Bethe ansatzaequations of the system. This is the asymptotic Bethe ansatz method.