Open Boundary From The Solution Of The Ladder Model

The integrable statistical model is a kind of important model in the statistical physics.It studies a simplified real physical system.Through the simplification ,the partition function of a complex system which is difficult to solve can be changed to easily solved one ,so the integrable model is very important and can arouse people's attention .On the other hand ,after we study it deeply,we find that it can be applied on many associated sub-jects,such as, in the mathematical physics, it directly lead to the appearance of the quantum group and some study methods used in it can be generalized to the condensed matter and the integrable field theory. Spin-ladder model is one of the most significant models of the integrable model.Although some people have solved the eigenstate and eigenvalue with periodic boundary condition ,the open boundary condition has not been given out yet .This is worth studying.In the second part and in the third part,the eigenvalue and the two-particle scattering matrix are obtained in studying the electron spin-ladder model under periodic boundary condition and under open boundary condition with coordinate BA method in integrable.The first ,the eigenvalue equation is formed according to schrodinger equation.The second ,the eigenvalue is worked out when the wave function is given .At the end ,the two-particle scattering matrix is calculated by eigenvalue equation and the continuity condition of the wave function.In the fourth part ,we will apply the nested BA method to find the eigenvalue and the eigenvector of the transfer matrix for spin-ladder model with open boundary conditions.First based on the reflection equation ,we find the general diagonal solution,which determine the open boundary interaction in the Hamiltonian .Next we introduce the spin-ladder model with open boundary conditions by finding the solution K+ of the reflection equation which determines the nontrivial boundary terms inthe Hamiltonian. At last, contributes to the diagonalization of the transfer matrix of the model with open boundary conditions in the frame work of nested BA .In the last the summary of our main results is presented .