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Journal of the Physical Society of Japan

Realization of Scattering Data Operators in Terms of Reflection Coefficient Operators in HeisenbergXXZModel

Yuji Nakawaki

Published Dec 15, 1990• 0 citations
Publisher

Abstract

We give an infinite volume formulation of quantum inverse scattering method for Heisenberg XXZ model, by making use of Jost solutions of difference equations satisfying prescribed asymptotic boundary conditions. We find operator identities among the Jost solutions which enable us to obtain Gel'fand-Levitan equations more easily than any previous formulations. By solving an integral equation, we express scattering data operators in terms of solely reflection coefficient operators, which are creation and annihilation operators of magnons and their bound states. We also see that when expressed in terms of them, Hamiltonian and other integrals of motion are diagonalized in free bosonic forms so that magnon and their bound states constitute a complete set of bases of Fock space.

DOI
10.1143/jpsj.59.4245

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We give an infinite volume formulation of quantum inverse scattering method for Heisenberg XXZ model, by making use of Jost solutions of difference equations satisfying prescribed asymptotic boundary conditions. We find operator identities among the Jost solutions which enable us to obtain Gel'fand-Levitan equations more easily than any previous formulations. By solving an integral equation, we express scattering data operators in terms of solely reflection coefficient operators, which are creation and annihilation operators of magnons and their bound states. We also see that when expressed in terms of them, Hamiltonian and other integrals of motion are diagonalized in free bosonic forms so that magnon and their bound states constitute a complete set of bases of Fock space.
We consider the nonlinear Schrödinger model of spin-21\frac{2}{1}12​ particles with attractive coupling to show a way to complete an infinite volume formulation of the quantum inverse scattering method. We use not any lattice models but only properties of the Jost functions satisfying given asymptotic boundary conditions in infinities. We find orthonormality relations of scattering data operators as well as underlying ones of the Jost functions and use them to derive operator identities satisfied by inverse operators of scattering data operators. Consequently we can simplyfy constructing the Gel'fand-Levitan equations. We also realize scattering data operators in terms of reflection coefficient operators.
The quantum generalization of the Gelfand-Levitan method is presented for the nonlinear Schr\"odinger model. The basic dispersion relation for operator Jost functions is derived, and the Heisenberg field operator is expressed in terms of scattering-data operators. Construction of Green's functions in the zero-density vacuum is discussed. The four-point function is explicitly calculated from the expression for the field operator and compared with the result of a direct Feynman graph summation. In addition it is proved for any number of particles that the Hamiltonian eigenstates constructed from the quantized scattering data are identical with those previously obtained by means of Bethe's ansatz.
The generalisation of the quantum inverse scattering method to the study of direct and inverse problems for the multicomponent non-linear Schrodinger model of bosons or fermions with repulsive coupling is made. Two sets of Yang-Baxter equations are solved to obtain commutation relations between the scattering state operators. The eigenfunctions have been constructed for the infinite number of conserved quantities and the eigenvalues of the first three conserved quantities-number of particles, momentum and energy-are obtained. The global Izergin-Korepin relations and the relations between the quantum Jost functions are derived, and from them, the quantum Gel'fand-Levitan equations are established. Finally, the series expansion for the field operators in terms of the scattering state operators is written out explicitly.
Subject of this work are integrable spin chains with general boundary conditions.In the framework of the Quantum Inverse Scattering Method Sklyanin has shown how to construct a family of commuting operators (transfer matrix) containing the hamiltonian of the XXX or XXZ spin chain with general boundary fields.Key ingredient is the underlying algebraic structure which is a combination of the Yang-Baxter algebra, using the known R-matrix representations, and a so-called Reflection algebra.The latter includes fields of arbitrary strength and direction acting on the first and last position of the chain.This setup is solvable via algebraic Bethe ansatz in the case of diagonal boundaries, i.e. the fields are parallel to each other and in the case of the XXZ model parallel to the distinguished direction.Kitanine et al. have managed to express local operators in terms of the non-local elements of the underlying algebraic structure in the case of half-infinite chain length hence establishing a possible approach to evaluate expectation values of physical observables.Their results will be picked up in this work and generalized to spin chains of arbitrary (including finite) lengths using non-linear integral equations for the lowest lying state with zero magnetization.In the case of non-diagonal boundary fields the lack of a reference state or pseudo vacuum prohibits the solution by algebraic Bethe ansatz.The method of separation of variables proposed by Sklyanin is not constrained in that sense and will be applied to this situation.In this approach for the XXX spin chain no restrictions to the boundary parameters are needed.The result is a TQ-equation on a finite discrete set of points and the eigenvalues of the transfer matrix are obtainable from this finite difference equation.As the underlying algebraic structure is independent of the representation, the analysis for the XXX spin chain can be extended to a spin-boson model.Using a known representation of the algebra for the bosonic degree of freedom again a TQ-equation is obtained but in this case being defined on an infinite discrete set of points.Finally a different approach to the case of non-diagonal boundary conditions using non-linear integral equations is studied.By construction of higher dimensional representations of the Yang Baxter algebra a hierarchy of transfer matrices is obtained.In this construction two R-matrices are fused together yielding the name of fusion hierarchy.From such hierarchies it is possible to derive non-linear integral equations for a certain eigenvalue of the transfer matrix using Fourier transform techniques.The equations are valid for arbitrary system sizes and a range of boundary parameters.By careful analysis of the zero and pole structure it is possible to extract information about the eigenvalue of a specific state.