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Physical review. B, Condensed matter

Integrable open-boundary conditions for the one-dimensional Bariev chain

Huan-Qiang Zhou

Published Mar 1, 1996• 9 citations
Publisher

Abstract

The integrable open-boundary conditions for the one-dimensional Bariev chain are considered. The diagonal boundary $K$ matrices are found and the commuting transfer matrix is constructed. Since the local monodromy matrix as well as the quantum $R$ matrix do not possess the crossing symmetry, our construction shows that Sklyanin's formalism may be extended to apply to any one-dimensional systems integrable by the quantum inverse scattering method.

DOI
10.1103/physrevb.53.5098

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The integrable open-boundary conditions for the one-dimensional Bariev chain are considered. The diagonal boundary KKK matrices are found and the commuting transfer matrix is constructed. Since the local monodromy matrix as well as the quantum RRR matrix do not possess the crossing symmetry, our construction shows that Sklyanin's formalism may be extended to apply to any one-dimensional systems integrable by the quantum inverse scattering method.
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.
We investigate the exact integrability of the one-dimensional (1D) Bariev model in the framework of the quantum inverse scattering method (QISM). Using the Jordan - Wigner transformation, the 1D Bariev model can be regarded as a coupled spin model. We construct the higher conserved currents which commute with the Hamiltonian. The explicit form of the conserved currents helps us to infer the L-operator of the 1D Bariev model. From the L-operator, we construct a transfer matrix which is a generating function of the conserved currents. We also find the corresponding R-matrix which satisfies the Yang - Baxter relation. Thus the exact integrability of the 1D Bariev model is established. The R-matrix does not have the `difference property' for the spectral parameter, as in the case of the 1D Hubbard model. We also provide the Lax representation and the fermionic formulation of the Yang - Baxter relation.
A simple relation between inhomogeneous transfer matrices and boundary quantum Knizhnik-Zamolodchikov (KZ) equations is exhibited for quantum integrable systems with reflecting boundary conditions, analogous to an observation by Gaudin for periodic systems. Thus, the boundary quantum KZ equations receive a new motivation. We also derive the commutativity of Sklyanin’s boundary transfer matrices by merely imposing appropriate reflection equations, in particular without using the conditions of crossing symmetry and unitarity of the R-matrix.
Integrability of strongly correlated electron systems on open chains XIANG-YU GEIn recent years, there has been a strong increase of interest in models of strongly correlated electron systems in condensed matter physics, in particular in integrable onedimensional models.Examples of such models are the one-dimensional Hubbard model, the supersymmetric t-J model, the Hubbard-like model (Bariev's chain), the so-called extended Hubbard model (EKS model), the supersymmetric U model and the eight-state supersymmetric U model.The study of integrable open-boundary conditions for strongly correlated electron models is very important for physical applications, especially for gaining insights into the behaviour of more realistic systems.The important ingredient in Sklyanin's boundary quantum inverse scattering method (QISM), which has played a key role in recent advances in the study of lattice models with open-boundaries, is a new algebraic structure, the reflection equation algebra.In this thesis, a general graded reflection equation algebra is proposed and the corresponding graded (supersymmetry) boundary QISM is formulated.The formalism is applicable to all boundary lattice systems where an invertible i?-matrix exists.As applications, some important integrable boundary conditions for various strongly correlated electron models with quantum superalgebra symmetries are constructed, including:1. Presentation of integrable open-boundary conditions for the one-dimensional qdeformed supersymmetric Essler-Korepin-Schoutens (EKS) extended Hubbard model.Determination of three different diagonal boundary if-matrices and of nine classes of integrable open-boundary conditions.Construction of the exactly solvable Hamiltonian with open-boundary terms for quantum superalgebra U q (gl(2\2)) symmetry; 2. Establishment of the quantum integrability of open-boundary conditions for the one-dimensional ^-deformed supersymmetric U model with quantum superalgebra U q (gl(2\l)} symmetry.Determination of diagonal boundary if-matrices and of integrable open-boundary conditions.Solution of the boundary system by means of the coordinate space Bethe ansatz technique, and derivation of the Bethe ansatz equations;