T-W Scheme In The One-dimensional Hubbard Model And Supersymmetric T-J Model

Quantum integrable models play an important role in a variety of fields such as condensed matter physics and statistical physics.For example,the exact solution of the one-dimensional(1D)Hubbard model with periodic boundary condition clarifies the physical nature of Mott insulator.Moreover,the physical picture of Luttinger liquid can be obtained from the exact solution of the supersymmetric t-J model with periodic boundary condition.It is fair to say that the study of quantum integrable systems is indispensable for understanding certain physical behaviors.Up to now there are two possible boundary conditions associated with quantum integrability:periodic boundary conditions and total reflecting boundary conditions.The study of the systems with boundary magnetic fields is of significant importance.However,if the boundary fields are unparallel,the non-diagonal boundary reflections will break the U(1)symmetry.Thus,in the T-Q scheme,the eigenvalues of the transfer matrix could only be parameterized in the form of inhomogeneous T-Q relation,and accordingly,the Bethe ansatz equations(BAEs)obtained from such T-Q relation will also be inhomogeneous.We note that although the BAEs with inhomogeneous terms can still provide the energy spectrum of the system,it is hard to calculate the physical quantities in the thermodynamic limit on the basis of thermodynamic Bethe ansatz when associated BAEs are inhomogeneous.As typical strongly correlated electronic systems,the 1D Hubbard model and the supersymmetric t-J model have been studied extensively,and some interesting results are obtained.However,at present,the exact thermodynamic analysis and some important physical properties including the ground state and the elementary excitations of these two models with unparallel boundary fields are still missing.In order to wipe out the inhomogeneous terms in the BAEs,in this dissertation,we will propose a novel parameterization scheme for the eigenvalues of the transfer matrix,namely the t-W scheme.This dissertation is organized as follows.In chapter 1,we introduce the basic concept of integrability.we provide integrable conditions for systems with two integrable boundary conditions,respectively,and introduce the most popular methods for solving quantum integrable models.In chapter 2 and chapter 3,we will apply the t-W parameterization,and solve the Hubbard model and the supersymmetric t-J model with different boundary conditions.During the process,we first construct the eigenvalue problem of the transfer matrix on the basis of coordinate Bethe ansatz,then we construct the t-W relation and obtain the corresponding BAEs which are all homogeneous.In chapter 4,we will take the Hubbard model with periodic boundary condition as an example,and introduce briefly the thermodynamic Bethe ansatz method,then we will point out the reason why this method can not be applied to systems where the associated BAEs are inhomogeneous.Finally,we will briefly analyze the points needed to be attended when calculating the thermodynamic quantities of the system in the t-W scheme.