Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1408.3364 · Journal of Mathematical Physics

Boundary transfer matrices and boundary quantum KZ equations

Bart Vlaar

Published Jul 1, 2015• 15 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.4927305

Related papers

Gaudin magnet with boundary and generalized Knizhnik-Zamolodchikov equation

Kazuhiro Hikami

We consider the boundary quantum inverse scattering method established by Sklyanin (1988). The Gaudin magnet with boundary is diagonalized by taking a quasi-classical limit of the inhomogeneous lattice. Using the method proposed by Babujian (1993), the integral representation for the solution of the B-type Knizhnik-Zamolodchikov equation is explicitly constructed.

Journal of Physics A Mathematical and General•Sep 7, 1995•44 citations

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

Huan-Qiang Zhou

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.

Physical review. B, Condensed matter•Mar 1, 1996•9 citations

Boundary Quantum Knizhnik-Zamolodchikov Equations and Integrability of Quantum Field Theories with Time-Dependent Bulk and Boundary Coupling Strengths

Parameshwar R. Pasnoori

The generalized Bethe ansatz framework formulated in [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)] provides a unified framework to find exact solutions to quantum many-body systems with time-dependent coupling strengths with periodic boundary conditions. In this work we extend this framework to the case of open boundary conditions where in addition to the time-dependent interactions in the bulk, the boundary conditions are explicitly time-dependent. We show that for integrable time-dependent bulk coupling strengths, the generalized Bethe ansatz framework provides the time-dependent boundary conditions compatible with integrability and reduces the time-dependent Schrodinger equation to a set of matrix difference equations called the boundary quantum Knizhnik-Zamolodchikov (BqKZ) equations. The solution to the BqKZ equations provides the explicit form of the exact wavefunction. We further show that the RG invariants of the corresponding static model identify with the dynamical invariants in the time-dependent model.

math-phCross-listedarXiv:2608.31080•Sep 1, 2026•0 citations

Truncation of Functional Relations in the XXZ Model

Alexander A. Belavin, S. Yu. Gubanov, Boris Lvovich Feigin

This is the abstract of the revised paper. The integrable XXZ model with a special open boundary condition is considered. We study Sklyanin transfer matrices after quantum group reduction in roots of unity. In this case Sklyanin transfer matrices satisfy a closed system of truncated functional equations. The algebraic reason for the truncation is found.The important role in proving of the result is performed by Zamolodchikov algebra introduced in the paper.

arXiv:hep-th/0008011•Moscow Mathematical Journal•Jan 1, 2001•7 citations

Classical and quantum integrable systems with boundaries

Yi-Xin Chen, Xudong Luo, Ke Wu

We study two-dimensional classically integrable field theory with independent boundary conditions at each end, and obtain three possible generating functions for integrals of motion when this model is an ultralocal one. Classically integrable boundary conditions can be found when solving boundary equations. In the quantum case, we also find that the unitarity condition of the quantum R -matrix is sufficient to construct commutative quantities with boundaries, and its reflection equations are obtained.

arXiv:hep-th/9808150•Journal of Physics A Mathematical and General•Jan 1, 1999•2 citations

GAUDIN MAGNET WITH IMPURITY AND ITS GENERALIZED KNIZHNIK–ZAMOLODCHIKOV EQUATION

A. LIMA-SANTOS, WAGNER UTIEL

This work is concerned with the formulation of the boundary quantum inverse scattering method for the xxz Gaudin magnet coupled to boundary impurities with arbitrary exchange constants. The Gaudin magnet is diagonalized by taking a quasi-classical limit of the inhomogeneous lattice. Using the method proposed by Babujian, the integral representation for the solution of the Knizhnik–Zamolodchikov equation is explictly constructed and its rational limit discussed.

arXiv:nlin/0504060•International Journal of Modern Physics B•Jun 8, 2006•7 citations
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.
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.
The generalized Bethe ansatz framework formulated in [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)] provides a unified framework to find exact solutions to quantum many-body systems with time-dependent coupling strengths with periodic boundary conditions. In this work we extend this framework to the case of open boundary conditions where in addition to the time-dependent interactions in the bulk, the boundary conditions are explicitly time-dependent. We show that for integrable time-dependent bulk coupling strengths, the generalized Bethe ansatz framework provides the time-dependent boundary conditions compatible with integrability and reduces the time-dependent Schrodinger equation to a set of matrix difference equations called the boundary quantum Knizhnik-Zamolodchikov (BqKZ) equations. The solution to the BqKZ equations provides the explicit form of the exact wavefunction. We further show that the RG invariants of the corresponding static model identify with the dynamical invariants in the time-dependent model.
We study two-dimensional classically integrable field theory with independent boundary conditions at each end, and obtain three possible generating functions for integrals of motion when this model is an ultralocal one. Classically integrable boundary conditions can be found when solving boundary equations. In the quantum case, we also find that the unitarity condition of the quantum R -matrix is sufficient to construct commutative quantities with boundaries, and its reflection equations are obtained.