The Study Of Separation Of Variables For 1D Exactly Solvable Models

Integrable models not only contain beautiful mathematical structures,but also provide a benchmark for important physical problems by rigorous solutions.Thus the integrable field has attracted wide attention of many physicists and mathematicians.Recently,integrable models without U（1）-symmetry have been solved exactly,which puts the research on integrable models into the current research hotspots.The method for solving such models is known as the off-diagonal Bethe ansatz（ODBA）method.It is very important to analyze the physical properties of the models based on the exact solution given by ODBA method.This paper mainly studies the thermodynamics,seperation of variables（SoV）basis and the representation of correlation function in the one-dimensional integrable models,including the Heisenberg spin chain model with the antiperiodic boundary condition and the Izergen-Korepin（IK）model.Using the ODBA method,we can get the inhomogeneous T-Q relation for the Heisen-berg spin chain model with the antiperiodic boundary condition.Numerical results show that the Bethe roots constrained by the inhomogeneous Bethe ansatz equations possess string struc-tures under certain conditions.Based on this fact,we study the ground state energy,energy gap and other physical quantities of the XXZ spin chain with antiperiodic boundary conditions.Using the ODBA and SoV methods,we give the determinant representation and its homoge-neous limits for the correlation functions for the XXZ model with the antiperiodic boundary condition for the first time.The basic ideas of the calculation for correlation functions for the integrable models without U（1）-symmetry is as follows.One should use the ODBA method to solve the spectrum problem for the integrable models,and then construct the SoV basis.By in-serting the transfer matrix into the norm of eigenstate and SoV basis,we can retrieve the Bethe states.Generally,the Bethe states are expressed by the elements of the monodromy matrix.By expanding the elements of the monodromy matrix in the SoV basis,we can obtain the explicit expression for the Bethe states in the SoV basis.Thus,the explicit expressions for the norm of Bethe states and form factors and correlation functions can be calculated exactly.Generally,these expressions possess a determinant representation.The R-matrix of the IK model corresponds to the simplest twisted affine algebra A₂^（2）.The IK model has played a fundamental role in quantum integrable models associated with algebras beyond A-type.The construction of separation of variables basis for the IK model is of important significance for the study of eigenstates,correlation functions and dynamic properties for the integrable models related with non A-type（super）algebras).Taking the IK model as an example,we give the separation of variables basis for the quantum integrable systems related to the non A-type（super）algebras.We expand the Bethe states of this model with the SoV basis and obtain the exact recursive relation for the Bethe states.The SoV basis for the Hubbard model is given in Appendix E.