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:1810.09048 · Nuclear Physics B

The spectrum of quantum-group-invariant transfer matrices

Rafael I. Nepomechie, Ana L. Retore

Published Nov 23, 2018• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1016/j.nuclphysb.2018.11.017

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

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

Analytical Bethe ansatz for a A2n-1(2), Bn(1), Cn(1), Dn(1)quantum-algebra-invariant open spin chains

Simone Artz, Luca Mezincescu, Rafael I. Nepomechie

We determine the eigenvalues of the transfer matrices for integrable open quantum spin chains which are associated with the affine Lie algebras A 2n-1 (2) , B n (1) , C n (1) , D n (1) , and which have the quantum-algebra invariance U q (C n ), U q (B n ), U q (C n ), U q (D n ), respectively.

arXiv:hep-th/9504085•Journal of Physics A Mathematical and General•Sep 21, 1995•42 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

Analytical Bethe Ansatz for quantum-algebra-invariant spin chains

Luca Mezincescu, Rafael I. Nepomechie

We have recently constructed a large class of open quantum spin chains which have quantum-algebra symmetry and which are integrable. We show here that these models can be exactly solved using a generalization of the analytical Bethe Ansatz (BA) method. In particular, we determine in this way the spectrum of the transfer matrices of the Uq[su(2)]-invariant spin chains associated with A1(1) and A2(2) in the fundamental representation. The quantum-algebra invariance of these models plays an essential role in obtaining these results. The BA equations for these open chains are “doubled” with respect to the BA equations for the corresponding closed chains.

arXiv:hep-th/9110050•Nuclear Physics B•Mar 1, 1992•113 citations

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

Rafael I Nepomechie, Rodrigo A. Pimenta

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.

arXiv:1805.10144•Journal of Physics A Mathematical and Theoretical•Aug 9, 2018•13 citations
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.
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.
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.
We have recently constructed a large class of open quantum spin chains which have quantum-algebra symmetry and which are integrable. We show here that these models can be exactly solved using a generalization of the analytical Bethe Ansatz (BA) method. In particular, we determine in this way the spectrum of the transfer matrices of the Uq[su(2)]-invariant spin chains associated with A1(1) and A2(2) in the fundamental representation. The quantum-algebra invariance of these models plays an essential role in obtaining these results. The BA equations for these open chains are “doubled” with respect to the BA equations for the corresponding closed chains.