Study On The Exact Solutions Of Some Strongly Correlated Systems And Their Thermodynamic Properties

In statistical physics, the partition function is the foundation for calculating the thermodynamics of the system. Once known the exact partition function, it is easy to get various physical quantities such as entropy, specific heat and magnetic susceptibility etc. Given a physical system, how to exactly calculate the partition function is quite complicated. Usually, the partition function could be obtained by using some approximations or numerical calculation, which could not give an analytic expression. In order to investigate the critical phenomena exactly, some special methods were founded to calculate analytically the energy spectra or the exact expression of the partition function for some strongly correlated systems. Furthermore it is possible to study the thermodynamical properties of systems. In this thesis, we will discuss in detail the exact solution of Bariev model and Hubbard model with fractional statistics and deformed XXZ model, meantime, discussing their thermodynamic property. The organization of present thesis is as following:The first chapter dedicated to review some basic conception of the exactly solvable models.The boundary effects is an important topic in condensed matter physics. The exact solving system with various boundary condition has become research focus to uncover the affection of boundary on the system. In the second chapter, we discuss the exact solution of the Hubbard model with fractional statistics under open boundary conditions. In this model, the deformation parameter depends on spins and coordinates of the particles obeying the fractional statistics, which gives rise to the change in the wave function of the system. The key point is to construct the wave function in different domains associated with deformed parameter. By means of the coordinate Bethe ansatz(CBA), the model bulk integrability requests that not only two-body scattering matrix satisfies the Yang-Baxter relation but also deformation parameter have a restrictive relation. Meanwhile, the scattering matrix S is also associated with deformed parameter. Under open boundary conditions, we obtain the Bethe ansatz equations and the energy spectrum and four classes of compatible with the integrability boundary fields.In the third chapter, we give the exact solution of the Bariev model with fractional statistics under periodic boundary condition and discuss the thermodynamic property for both repulsive and attractive interactions. Using the CBA method, we obtain the Bethe ansatz equation and energy spectrum of the Hamiltonian of Bariev model with fractional statistics. Due to the fractional statistics, the wave functions in different regions connect with deformed parameter. Through calculations and analysis, we obtain a sets of constrain conditions for deformed parameters to be satisfied. Considering fractional statistics, we restrict the coordinate to a sector of ordered coordinates. However, the periodicity means the result being independent on where the starting point was set. These two points result the wave function phases shift, and diagonalizing the spin degree of freedom of the Hamiltonian is different from that before. We also derive out the thermodynamic Bethe ansatz equations and free energy based on the string hypothesis for both repulsive and attractive interactions. These equations are discussed in some limiting cases, such as the ground state, weak and strong coupling.In the fourth chapter, we constructed a new type of deformed XXZ model. Through the well-defined ansatz of the wave function, we diagonalized the Hamiltonian of the system by coordinate Bethe ansatz method. We obtained the energy and the Bethe ansatz equations of the model and also discussed some thermodynamics of the model.Gaudin system is a new quantum integrability system, The integrability of Gaudin model is related to classical r matrices of simple Lie algebras and semi-simple Lie algebra. In the fifth chapter, we construct the elliptic L{u) operator based on the classical Lie algebra A_n, B_n, C_n, D_n. With the help of the knowledge of elliptic function, We prove that the linear-poisson-lie brackets related to the given Lax matrix L(X) gives a r-matrices structure. Meanwhile, we present the generating functions of integrals of motion t(u). So that the function t(u) generales a family of commuting hamiltonian, system is integral.