Journal of the Physical Society of Japan

Supersymmetric Extension of the Reflection Equation Algebra and Integrable Boundary Conditions in Doped Spin-1 Model

Xiangyu Ge, Miki Wadati

Published Dec 11, 2006• 0 citations

Abstract

The reflection equation algebra of Sklyanin is extended to the supersymmetric case. A graded reflection equation algebra is proposed and the corresponding graded (supersymmetric) boundary quantum inverse scattering method (QISM) is formulated. As an application, integrable open-boundary conditions for the doped spin-1 chain of the supersymmetric t – J model are studied in the framework of the boundary QISM. Diagonal boundary K -matrices are found and four classes of integrable boundary terms are determined.

DOI: 10.1143/jpsj.75.124004

Related papers

Integrable Boundary Conditions for the One-Dimensional Hubbard Model

Masahiro Shiroishi, Miki Wadati

We discuss the integrable boundary conditions for the one-dimensional (1D) Hubbard Model in the framework of the Quantum Inverse Scattering Method (QISM). We use the fermionic R-matrix proposed by Olmedilla et al. to treat the twisted periodic boundary condition and the open boundary condition. We determine the most general form of the integrable twisted periodic boundary condition by considering the symmetry matrix of the fermionic R-matrix. To find the integrable open boundary condition, we shall solve the graded reflection equation, and find there are two diagonal solutions, which correspond to a) the boundary chemical potential and b) the boundary magnetic field. Non-diagonal solutions are obtained using the symmetry matrix of the fermionic R-matrix and the covariance property of the graded reflection equation. They can be interpreted as the SO(4) rotations of the diagonal solutions.

arXiv:cond-mat/9708011Journal of the Physical Society of JapanAug 15, 199762 citations

Lax pair formulation for the open boundary Osp(1∣2) spin chain

Xiaoyu Zhang, Kun Hao

Abstract Based on the Lax pair formulation, we study the integrable conditions of the Osp(1∣2) spin chain with open boundaries. We consider both the non-graded and graded versions of the Osp(1∣2) chain. The Lax pair operators M ± for the boundaries can be induced by the Lax operator M j for the bulk of the system. They correspond to the reflection equations (RE) and the Yang–Baxter equation, respectively. We further calculate the boundary K-matrices for both the non-graded and graded versions of the model with open boundaries. The double row monodromy matrix and transfer matrix of the spin chain have also been constructed. The K-matrices obtained from the Lax pair formulation are consistent with those from Sklyanin’s RE. This construction is another way to prove the quantum integrability of the Osp(1∣2) chain. We find that the Lax pair formulation has advantages in dealing with the boundary terms of the supersymmetric model.

Communications in Theoretical PhysicsSep 1, 20220 citations

Integrability of strongly correlated electron systems on open chains

Xiangyu Ge

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;

Bulletin of the Australian Mathematical SocietyJun 1, 20010 citations

Integrable Kondo impurities in the one-dimensional supersymmetric extended Hubbard model

Huan-Qiang Zhou, Xiangyu Ge, M. D. Gould

An integrable Kondo problem in the one-dimensional supersymmetric extended Hubbard model is studied by means of the boundary graded quantum inverse scattering method. The boundary K -matrices depending on the local moments of the impurities are presented as a non-trivial realization of the graded reflection equation algebras in a two-dimensional impurity Hilbert space. Further, the model is solved by using the algebraic Bethe ansatz method and the Bethe ansatz equations are obtained.

arXiv:cond-mat/9811048Journal of Physics A Mathematical and GeneralJan 1, 19995 citations

Non-diagonal boundary conditions for \mathfrak {gl}(1|1) super spin chains

André M Grabinski, Holger Frahm

We study a one-dimensional model of free fermions with supersymmetry and demonstrate how non-diagonal boundary conditions can be incorporated into the framework of the graded quantum inverse scattering method (gQISM) by means of super matrices with entries from a superalgebra. For super Hermitian twists and open boundary conditions subject to a certain constraint, we solve the eigenvalue problem for the super transfermatrix by means of the graded algebraic Bethe ansatz technique (gABA) starting from a fermionic coherent state. For generic boundary conditions the algebraic Bethe ansatz cannot be applied. In this case the spectrum of the super transfermatrix is obtained from a functional relation.

arXiv:0910.4029Journal of Physics A Mathematical and TheoreticalJan 4, 201016 citations

On the Quantum Integrability for A Model of Coupled XY Chains

Huan-Qiang Zhou, Xiangyu Ge

The paper discusses integrable open-boundary conditions for the model of three coupled one-dimensional XY spin chains in the framework of the quantum inverse scattering method. The diagonal boundary K-matrices are found and a class of integrable boundary terms are determined. The boundary model Hamiltonian is solved by using the coordinate space Bethe ansatz technique and Bathe ansatz equations are derived.

Jan 1, 19990 citations