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:1805.10144 · Journal of Physics A Mathematical and Theoretical

New D n + 1 ( 2 ) K-matrices with quantum group symmetry

Rafael I Nepomechie, Rodrigo A. Pimenta

Published Aug 9, 2018• 13 citations
arXivHTMLPDF

Abstract

Abstract We propose new families of solutions of the boundary Yang–Baxter equation. The open spin-chain transfer matrices constructed with these K-matrices have quantum group symmetry corresponding to removing one node from the Dynkin diagram, namely, , where . These transfer matrices also have a duality symmetry. These symmetries help to account for the degeneracies in the spectrum of the transfer matrix.

DOI
10.1088/1751-8121/aad957

Related papers

Surveying the quantum group symmetries of integrable open spin chains

Rafael I. Nepomechie, Ana L. Retore

Using anisotropic R-matrices associated with affine Lie algebras gˆ (specifically, A2n(2), A2n−1(2), Bn(1), Cn(1), Dn(1)) and suitable corresponding K-matrices, we construct families of integrable open quantum spin chains of finite length, whose transfer matrices are invariant under the quantum group corresponding to removing one node from the Dynkin diagram of gˆ. We show that these transfer matrices also have a duality symmetry (for the cases Cn(1) and Dn(1)) and additional Z2 symmetries that map complex representations to their conjugates (for the cases A2n−1(2), Bn(1) and Dn(1)). A key simplification is achieved by working in a certain “unitary” gauge, in which only the unbroken symmetry generators appear. The proofs of these symmetries rely on some new properties of the R-matrices. We use these symmetries to explain the degeneracies of the transfer matrices.

arXiv:1802.04864•Nuclear Physics B•Mar 5, 2018•15 citations

The spectrum of quantum-group-invariant transfer matrices

Rafael I. Nepomechie, Ana L. Retore

Integrable open quantum spin-chain transfer matrices constructed from trigonometric R-matrices associated to affine Lie algebras g ˆ , and from certain K-matrices (reflection matrices) depending on a discrete parameter p , were recently considered in arXiv:1802.04864 and arXiv:1805.10144 . It was shown there that these transfer matrices have quantum group symmetry corresponding to removing the p th node from the g ˆ Dynkin diagram. Here we determine the spectrum of these transfer matrices by using analytical Bethe ansatz, and we determine the dependence of the corresponding Bethe equations on p . We propose formulas for the Dynkin labels of the Bethe states in terms of the numbers of Bethe roots of each type. We also briefly study how duality transformations are implemented on the Bethe ansatz solutions.

arXiv:1810.09048•Nuclear Physics B•Nov 23, 2018•9 citations

Nonstandard coproducts and the Izergin-Korepin open spin chain

Rafael I. Nepomechie

Corresponding to the Izergin-Korepin ( A 2 (2) ) R matrix, there are three diagonal solutions (` K matrices') of the boundary Yang-Baxter equation. Using these R and K matrices, one can construct transfer matrices for open integrable quantum spin chains. The transfer matrix corresponding to the identity matrix K = is known to have U q ( o (3)) symmetry. We argue here that the transfer matrices corresponding to the other two K matrices also have U q ( o (3)) symmetry, but with a nonstandard coproduct. We briefly explore some of the consequences of this symmetry.

arXiv:hep-th/9911232•Journal of Physics A Mathematical and General•Dec 23, 1999•12 citations

A Q-Operator for Open Spin Chains II: Boundary Factorization

Alec Cooper, Bart Vlaar, Robert Weston

Abstract One of the features of Baxter’s Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang–Baxter equation associated to particular infinite-dimensional representations). To extend such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang–Baxter equation) associated to these representations. In the case of quantum affine $$\mathfrak {sl}_2$$ sl 2 and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.

arXiv:2301.03997•Communications in Mathematical Physics•Apr 25, 2024•3 citations

Quantum group symmetries and completeness for A 2 n ( 2 ) open spin chains

Ibrahim Ahmed, Rafael I. Nepomechie, Wang Chunguang

Abstract We argue that the Hamiltonians for A 2 n ( 2 ) open quantum spin chains corresponding to two choices of integrable boundary conditions have the symmetries U q ( B n ) and U q ( C n ) , respectively. We find a formula for the Dynkin labels of the Bethe states (which determine the degeneracies of the corresponding eigenvalues) in terms of the numbers of Bethe roots of each type. With the help of this formula, we verify numerically (for a generic value of the anisotropy parameter) that the degeneracies and multiplicities of the spectra implied by the quantum group symmetries are completely described by the Bethe ansatz.

arXiv:1702.01482•Journal of Physics A Mathematical and Theoretical•May 31, 2017•5 citations

The integrable quantum group invariant A2n−1(2) and Dn+1(2) open spin chains

Rafael I. Nepomechie, Rodrigo A. Pimenta, Ana L. Retore

A family of A2n(2) integrable open spin chains with Uq(Cn) symmetry was recently identified in arXiv:1702.01482. We identify here in a similar way a family of A2n−1(2) integrable open spin chains with Uq(Dn) symmetry, and two families of Dn+1(2) integrable open spin chains with Uq(Bn) symmetry. We discuss the consequences of these symmetries for the degeneracies and multiplicities of the spectrum. We propose Bethe ansatz solutions for two of these models, whose completeness we check numerically for small values of n and chain length N. We find formulas for the Dynkin labels in terms of the numbers of Bethe roots of each type, which are useful for determining the corresponding degeneracies. In an appendix, we briefly consider Dn+1(2) chains with other integrable boundary conditions, which do not have quantum group symmetry.

arXiv:1707.09260•Nuclear Physics B•Sep 7, 2017•14 citations
Abstract We propose new families of solutions of the boundary Yang–Baxter equation. The open spin-chain transfer matrices constructed with these K-matrices have quantum group symmetry corresponding to removing one node from the Dynkin diagram, namely, , where . These transfer matrices also have a duality symmetry. These symmetries help to account for the degeneracies in the spectrum of the transfer matrix.
Using anisotropic R-matrices associated with affine Lie algebras gˆ (specifically, A2n(2), A2n−1(2), Bn(1), Cn(1), Dn(1)) and suitable corresponding K-matrices, we construct families of integrable open quantum spin chains of finite length, whose transfer matrices are invariant under the quantum group corresponding to removing one node from the Dynkin diagram of gˆ. We show that these transfer matrices also have a duality symmetry (for the cases Cn(1) and Dn(1)) and additional Z2 symmetries that map complex representations to their conjugates (for the cases A2n−1(2), Bn(1) and Dn(1)). A key simplification is achieved by working in a certain “unitary” gauge, in which only the unbroken symmetry generators appear. The proofs of these symmetries rely on some new properties of the R-matrices. We use these symmetries to explain the degeneracies of the transfer matrices.
Integrable open quantum spin-chain transfer matrices constructed from trigonometric R-matrices associated to affine Lie algebras g ˆ , and from certain K-matrices (reflection matrices) depending on a discrete parameter p , were recently considered in arXiv:1802.04864 and arXiv:1805.10144 . It was shown there that these transfer matrices have quantum group symmetry corresponding to removing the p th node from the g ˆ Dynkin diagram. Here we determine the spectrum of these transfer matrices by using analytical Bethe ansatz, and we determine the dependence of the corresponding Bethe equations on p . We propose formulas for the Dynkin labels of the Bethe states in terms of the numbers of Bethe roots of each type. We also briefly study how duality transformations are implemented on the Bethe ansatz solutions.
Corresponding to the Izergin-Korepin ( A 2 (2) ) R matrix, there are three diagonal solutions (` K matrices') of the boundary Yang-Baxter equation. Using these R and K matrices, one can construct transfer matrices for open integrable quantum spin chains. The transfer matrix corresponding to the identity matrix K = is known to have U q ( o (3)) symmetry. We argue here that the transfer matrices corresponding to the other two K matrices also have U q ( o (3)) symmetry, but with a nonstandard coproduct. We briefly explore some of the consequences of this symmetry.
Abstract One of the features of Baxter’s Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang–Baxter equation associated to particular infinite-dimensional representations). To extend such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang–Baxter equation) associated to these representations. In the case of quantum affine sl2\mathfrak {sl}_2sl2​ sl 2 and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.
Abstract We argue that the Hamiltonians for A 2 n ( 2 ) open quantum spin chains corresponding to two choices of integrable boundary conditions have the symmetries U q ( B n ) and U q ( C n ) , respectively. We find a formula for the Dynkin labels of the Bethe states (which determine the degeneracies of the corresponding eigenvalues) in terms of the numbers of Bethe roots of each type. With the help of this formula, we verify numerically (for a generic value of the anisotropy parameter) that the degeneracies and multiplicities of the spectra implied by the quantum group symmetries are completely described by the Bethe ansatz.
A family of A2n(2) integrable open spin chains with Uq(Cn) symmetry was recently identified in arXiv:1702.01482. We identify here in a similar way a family of A2n−1(2) integrable open spin chains with Uq(Dn) symmetry, and two families of Dn+1(2) integrable open spin chains with Uq(Bn) symmetry. We discuss the consequences of these symmetries for the degeneracies and multiplicities of the spectrum. We propose Bethe ansatz solutions for two of these models, whose completeness we check numerically for small values of n and chain length N. We find formulas for the Dynkin labels in terms of the numbers of Bethe roots of each type, which are useful for determining the corresponding degeneracies. In an appendix, we briefly consider Dn+1(2) chains with other integrable boundary conditions, which do not have quantum group symmetry.