Bethe Ansatz Solutions Of The Quantum Integrable Model With Off-diagonal Boundary Conditions

Quantum integrable model is a special non-linear quantum many-body system in the field of mathematical physics.The precise results of these models can provide a solid benchmarks for many important physical problems.In recent decades,the major difficulty of solving quantum integrable model lies in that we can’t obtain the spectrum and the wave function of the U（1）symmetry broken system by the algebra Bethe anstz method and the traditional T-Q method.To this end,Yupeng Wang and his collaborators developed a new analytic method–the Off-diagonal Bethe Ansatz method.The innovation point of this method is to plus the inhomogeneous term in the traditional T-Q relations.That gives the universal inhomogeneous T-Q relations.Based on this method,we mainly studied the exact solution of several U（1）symmetry broken quantum integrable models in different boundary conditions.The details are as follows:Bethe ansatz solutions of the small polaron with off-diagonal boundary conditionsThe small polaron model is a typical spinless fermion model which provides an effective description of the motion of an additional electron in a polar crystal.It plays an important role in low dimensional condensed matter physics.For the small polaron model with off-diagonal boundary conditions,the Hamiltonian includes Grassmann valued non-diagonal boundary fields which breaks the bulk U（1）-symmetry of the model.The traditional method is useless for this model because we are not able to find out the obvious reference state（all-spin-up or all-spindown state）.However,the new proposed off-diagonal Bethe ansatz method is very effective and universal tool for solving U（1）-symmetry breaking quantum integral models.Therefore,we use the fundamental off-diagonal Bethe ansatz method to obtain the eigenvalue of the Hamilton and the associated Bethe ansatz equations for the small polaron model with off-diagonal boundary conditions.Bethe ansatz solutions of the τ₂-model with diagonal boundary conditionsτ₂-model is one of the simplest quantum integrable models associated with cyclic representation of the Weyl abgebra.It under certain parameter constraint is highly related to some other integrable models.In particular,Many efforts have been made to obtain the solutions of chiral Potts model by solving the τ₂-model with a recursive functional relation.However,it was found that only in the special case the traditional method can be applied on this model.Therefore,for the general solutions of the τ₂-model with diagonal boundary,we obtained the eigenvalue of the corresponding transfer matrix of the model and the associated Bethe ansatz equations by the fusion procedure and the off-diagonal Bethe ansatz method.Also,we give the numerical solutions of the Bethe ansatz equations when the lattice number N is small,which imply that the inhomogeneous T-Q relation does indeed give the complete and correct spectrum of the generic τ₂transfer matrix.In addition,we get the further constraints that inhomogeneity parameters satisfy when the inhomogeneous T-Q relations reduces to the conventional one.Bethe ansatz solutions of the τ₂-model with generic open boundary conditionsFinally,we studied the exact solutions of the τ₂-model in general open boundary conditions.Similarly,Based on the truncation identity of the fused transfer matrices and the asymptotic behaviors of the fundamental transfer matrix,we combine the fusion procedure and use the off-diagonal Bethe ansatz method to obtain the eigenvalue of the fundamental transfer matrix by constructing the inhomogeneous T-Q relation,the related Bethe ansatz equations and the degradation conditions.