The Study Of Exact Solution Of Quantum Spin Chain With Different Boundary Conditions

In this article,we introduce the exact solutions of one-dimensional quantum spin chain with several different boundary conditions.In the first chapter,we firstly introduce the Bethe ansatz method,which plays an important role in the quantum integrable systems.We also mention some different forms of the Bethe ansatz method,such as algebraic Bethe ansatz,nested Bethe ansatz and off-diagonal Bethe ansatz method as well as the related concepts and techniques in the application of these methods.At the end of this chapter,we introduce how to get the thermodynamic properties of the system by the Bethe ansatz equations.In the second chapter,we construct an operator solution of the reflection equation.Based on it,we propose an integrable spin-1 chain with a magnetic impurity whose spin is 1/2.Due to the existence of impurity,the SU(3)symmetry of the bulk is broken.By using the nested algebraic Bethe ansatz,we obtain the exact solution of the system.The eigenvalues,eigenstates and Bethe ansatz equations are given explicitly.In the third chapter,we construct an integrable quantum spin chain which includes the nearest-neighbor,next-nearest-neighbor,chiral three-spin couplings,DzyloshinskyMoriya interactions and unparallel boundary magnetic fields.Although the interaction in the bulk are isotropic,the spins nearby the boundary fields are polarized which induce the anisotropic exchanging interactions of the first and last bonds.The U(1)symmetry of the system is broken because of the off-diagonal boundary reflections.By using the off-diagonal Bethe ansatz method,we obtains the exact solution of the system.The inhomogeneous T-Q relation and the Bethe ansatz equations are given explicitly.In the forth chapter,we get the exact solution of an integrable anisotropic Heisenberg spin chain.It has nearest-neighbour interactions,next-nearest-neighbour interactions and scalar chirality coupling as well as the antiperiodic boundary conditions.The detailed construction of the Hamiltonian and the proof of integrabilily are given.The antiperiodic boundary condition breaks the U(1)-symmetry of the system and we use the off-diagonal Bethe ansatz to solve it.The energy spectrum is characterized by the inhomogeneous T-Q relations and the contributes of the inhomogeneous term is studied.The ground state energy and the twisted boundary energy in different regions are otained.We also find that the Bethe roots at the ground state form the string structure if the coupling constant J=-1 although the Bethe anstaz equations are the inhomogeneous ones.