The Study Of Bethe States And Surface Energy Of 1D Exactly Solvable Models

The quantum integrable models played a pivotal role in providing concrete realizations of dif-ferent phenomenologies in many fields such as quantum field theory,condensed matter physics and statistical physics.For the integrable models with U(1)symmetry,the eigenvalues and eigenstates can be given successfully by the coordinate Bethe ansatz and the algebraic Bethe ansatz method.For the integrable models without U(1)symmetry,the off-diagonal Bethe ansatz is very useful.Based on the inhomogeneous T-Q relation constructed via the off-diagonal Bethe Ansatz,a systematic method for retrieving the Bethe-type eigenstates(Bethe states)of integrable models without obvious reference state is also developed by employing certain or-thogonal basis of the Hilbert space.In this paper we mainly study the eigenstates and the surface energy of the one dimensional integrable models with general boundary conditions,including the trigonometric SU(3)spin chain,Heisenberg spin chain and one-dimensional supersymmet-ric t-J model.The t-J model can be used to calculate high temperature superconductivity states.XXZ model is very important in studying the quantum phase transition.For the trigonometric SU(3)spin chain,by using the algebraic Bethe ansatz,we found that the eigenstates can be reduced to the Bethe states problem of the XXZ spin chain.Then,by using the off-diagonal Bethe ansatz,we can give the inhomogeneous T-Q relation,the corresponding eigenstates are expressed in terms of nested Bethe-type eigenstates which have well-defined homogeneous limit.In the end,numerical results for the small size systems show that the spectrum and the Bethe states obtained by the nested Bethe ansatz equations are exactly correct.For the one-dimensional supersymmetric t-J model with generic open boundaries,first,we introduced the associated graded R-matrix and the corresponding generic integral non-diagonal boundary reflection matrices;then,by using the graded algebraic Bethe ansatz,we found that the eigenstates of the transfer matrix can be expressed by the Bethe states of the nested transfer matrix.The Bethe states of the nested transfer matrix are derived by using the off-diagonal Bethe ansatz;in the end,with the help of the Density Matrix Renomalization Group(DMRG),we found that the contribution of the inhomogeneous term to the surface energy can be neglect-ed when the system-size L tends to infinity.For the spin-₂¹XXZ chain with arbitrary boundary fields,in the thermodynamic limit,we study the ground state energy,elementary excitation and the surface energy.By using the Yang-Yang method and DMRG,we found that the contribution of the inhomogeneous term can be neglect-ed when the system-size N tends to infinity.Based on the reduced Bethe ansatz equation,we study the surface energy which contains the effects induced by the unparallel boundary fields.We show that how to construct the Bethe states of some integrable models with general bound-ary conditions.And we also calculated the the contribution of the inhomogeneous term to the surface energy when the system-size N tends to infinity.This is very important for further study of thermodynamics.