The Eigenvalue And Eigenstate Of Quantum Integrable Models Without Particle Number Conservation

This paper focuses on the quantum integrable models,which are essential in mathematics and Physics.In order to solve the eigenvalue problem of quantum integrable models and retrieve the Bethe states,we introduce and use several most popular methods:coordinate Bethe Ansatz,algebraic Bethe Ansatz,the T-Q relation proposed by Baxter,separation of variables and off-diagonal Bethe Ansatz.In the first part of this paper,we simply introduce the integrability,YangBaxter equation,reflection equation,quantum integrable models and several classical methods.In the second part of this paper,we study the antiperiodic XXZ spin chain,open XXX spin chain and open XXZ spin chain and develop a method to retrieve the Bethe states based on the inhomogenous T-Q relation and SoV basis.The process of retrieving the Bethe states is as follow: first,we need to construct the inhomogenous T-Q relation and give the corresponding Bethe Ansatz equations via the off-diagonal Bethe Ansatz method;then,we need to use the SoV method to construct a complete basis of the Hilbert space of the system,which is the eigenstates of the operator X(u);then,we calculate a set of scalar product of the given SoV basis and the eigenstate of transfer matrix,which can determine the eigenstate of transfer matrix completely;in the end,we construct the Bethe state with the help of the operators {X(uj)} and a proper reference state and then prove that the Bethe state is an eigenstate of the transfer matrix via the calculated scalar products.The reference state in the Bethe state of XXZ spin torus is a highly entangled superposition state and the creation operator X(uj)is an off-diagonal element of the monodromy matrix.The Bethe states of open XXX spin chain and open XXZ spin chain have the similar structure and we use two or two set of gauge transformations to obtain the operator and reference state for constructing Bethe vectors.The final result shows that the reference state is given by the triangularization of K-matrix and the creation operators are given by the diagonalization of K+matrix.In the third part of the paper,we give the exact solution of the onedimensional super-symmetric t-J model with unparallel boundary fields and an AdS/CFT open spin chain with non-diagonal boundaries respectively.With the coordinate Bethe Ansatz or algebraic Bethe Ansatz method,the eigenvalue problem of these two models are transformed into the eigenvalue problem of open spin chains with unparallel boundary fields,which are exactly solved via off-diagonal Bethe Ansatz method.With the help of the off-diagonal Bethe Ansatz solution,we firstly give the exact solution of these two non-trivial models.