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

An algebraic 21-vertex intertwiner for a non-elliptic XYZ Lax operator

Weiqi Jiang

Published Aug 23, 2026• 0 citations
Publisher

Abstract

Fukai and Yamada introduced a Lax operator with a three-dimensional auxiliary space for the spin-1/2 XYZ chain whose entries are algebraic rather than elliptic. We give an explicit algebraic 9 × 9 intertwiner for this operator. The matrix has the 21-vertex pattern and is written directly in terms of the two spectral parameters and the three XYZ couplings. Direct reduction in the radical function field proves the local RLL relation. We also prove an exchange-unitarity identity, generic uniqueness up to scalar, and regularity after a scalar renormalization. Its invariant blocks give a factorization of the characteristic polynomial and show that, on the common finite chart, the rank-loss locus is exactly 1 − R² = 0. We make the constant gauge to the fused eight-vertex matrix explicit and obtain regular XXZ and scaled rational XXX limits. The result supplies the local intertwiner left implicit by the fusion argument; it does not constitute a new integrable model.

DOI
10.5281/zenodo.22066996

Related papers

Relativistic classical integrable tops and quantum R-matrices

Andrei Mikhailovich Levin, Mikhail Aronovich Olshanetsky, Andrei Vladimirovich Zotov

We describe classical top-like integrable systems arising from the quantum exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed in terms of the quantum non-dynamical R-matrix even at the classical level, where the Planck constant plays the role of the relativistic deformation parameter in the sense of Ruijsenaars and Schneider (RS). The integrable systems (relativistic tops) are described as multidimensional Euler tops, and the inertia tensors are written in terms of the quantum and classical R-matrices. A particular case of gl N system is gauge equivalent to the N-particle RS model while a generic top is related to the spin generalization of the RS model. The simple relation between quantum R-matrices and classical Lax operators is exploited in two ways. In the elliptic case we use the Belavin’s quantum R-matrix to describe the relativistic classical tops. Also by the passage to the noncommutative torus we study the large N limit corresponding to the relativistic version of the nonlocal 2d elliptic hydrodynamics. Conversely, in the rational case we obtain a new gl N quantum rational non-dynamical R-matrix via the relativistic top, which we get in a different way — using the factorized form of the RS Lax operator and the classical Symplectic Hecke (gauge) transformation. In particular case of gl2 the quantum rational R-matrix is 11-vertex. It was previously found by Cherednik. At last, we describe the integrable spin chains and Gaudin models related to the obtained R-matrix.

arXiv:1405.7523•Journal of High Energy Physics•Jul 1, 2014•53 citations

Embedding integrable spin models in solvable vertex models on the square lattice

Marcio Jose Martins

Exploring a mapping among n -state spin and vertex models on the square lattice, we argue that a given integrable spin model with edge weights satisfying the rapidity difference property can be formulated in the framework of an equivalent solvable vertex model. The Lax operator and the R-matrix associated to the vertex model are built in terms of the edge weights of the spin model and these operators are shown to satisfy the Yang-Baxter algebra. The unitarity of the R-matrix follows from an assumption that the vertical edge weights of the spin model satisfy certain local identities known as inversion relation. We apply this embedding to the scalar n -state Potts model and we argue that the corresponding R-matrix can be written in terms of the underlying Temperley-Lieb operators. We also consider our construction for the integrable Ashkin-Teller model and the respective R-matrix is expressed in terms of sixteen distinct weights parametrized by theta functions. We comment on the possible extension of our results to spin models whose edge weights are not expressible in terms of the difference of spectral parameters.

arXiv:2501.02377•Nuclear Physics B•Feb 20, 2025•3 citations

Auxiliary matrices for the six-vertex model atqN 1 and a geometric interpretation of its symmetries

Christian Korff

The construction of auxiliary matrices for the six-vertex model at a root of unity is investigated from a quantum group theoretic point of view. Employing the concept of intertwiners associated with the quantum loop algebra U q ( 2 ) at q N = 1, a three-parameter family of auxiliary matrices is constructed. The elements of this family satisfy a functional relation with the transfer matrix allowing one to solve the eigenvalue problem of the model and to derive the Bethe ansatz equations. This functional relation is obtained from the decomposition of a tensor product of evaluation representations and involves auxiliary matrices with different parameters. Because of this dependence on additional parameters, the auxiliary matrices break in general the finite symmetries of the six-vertex model, such as spin-reversal or spin-conservation. More importantly, they also lift the extra degeneracies of the transfer matrix due to the loop symmetry present at rational coupling values. The extra parameters in the auxiliary matrices are shown to be directly related to the elements in the enlarged centre Z of the algebra U q ( 2 ) at q N = 1. This connection provides a geometric interpretation of the enhanced symmetry of the six-vertex model at rational coupling. The parameters labelling the auxiliary matrices can be interpreted as coordinates on a hypersurface Spec Z ⊂ ℂ 4 which remains invariant under the action of an infinite-dimensional group G of analytic transformations, called the quantum coadjoint action.

arXiv:math-ph/0302002•Journal of Physics A Mathematical and General•Apr 29, 2003•37 citations

Construction of some missing eigenvectors of the XYZ spin chain at the discrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix

Tetsuo Deguchi

We discuss an algebraic method for constructing eigenvectors of the transfer matrix of the eight-vertex model at the discrete coupling parameters. We consider the algebraic Bethe ansatz of the elliptic quantum group E τ,η ( sl 2 ) for the case where the parameter η satisfies 2 N η = m 1 + m 2 τ for arbitrary integers N , m 1 and m 2 . When m 1 or m 2 is odd, the eigenvectors thus obtained have not been discussed previously. Furthermore, we construct a family of degenerate eigenvectors of the XYZ spin chain, some of which are shown to be related to the sl 2 loop algebra symmetry of the XXZ spin chain. We show that the dimension of some degenerate eigenspace of the XYZ spin chain on L sites is given by N 2 L/N , if L/N is an even integer. The construction of eigenvectors of the transfer matrices of some related interaction round a face models is also discussed.

arXiv:cond-mat/0109078•Journal of Physics A Mathematical and General•Jan 21, 2002•33 citations

XYZ Integrability the Easy Way

Paul Fendley, Sascha Gehrmann, Eric Vernier +1 more

Abstract Sutherland showed that the XYZ quantum spin-chain Hamiltonian commutes with the eight-vertex model transfer matrix, so that Baxter’s subsequent tour de force proves the integrability of both. The proof requires parametrising the Boltzmann weights using elliptic theta functions and showing they satisfy the Yang-Baxter equation. We here give a simpler derivation of the integrability of the XYZ chain by explicitly constructing an extensive sequence of conserved charges from a matrix-product operator. We show that they commute with the XYZ Hamiltonian with periodic boundary conditions or an arbitrary boundary magnetic field. A straightforward generalisation yields impurity interactions that preserve the integrability. Placing such an impurity at the edge gives an integrable generalisation of the Kondo problem with a gapped bulk. We make contact with the traditional approach by relating our matrix-product operator to products of the eight-vertex model transfer matrix.

arXiv:2511.04674•Journal of Statistical Physics•Jun 19, 2026•1 citation

Characteristic determinant and Manakov triple for the double elliptic integrable system

A. Grekov, Andrei Zotov

Using the intertwining matrix of the IRF-Vertex correspondence we propose a determinant representation for the generating function of the commuting Hamiltonians of the double elliptic integrable system. More precisely, it is a ratio of the normally ordered determinants, which turns into a single determinant in the classical case. With its help we reproduce the recently suggested expression for the eigenvalues of the Hamiltonians for the dual to elliptic Ruijsenaars model. Next, we study the classical counterpart of our construction, which gives expression for the spectral curve and the corresponding L-matrix. This matrix is obtained explicitly as a weighted average of the Ruijsenaars and/or Sklyanin type Lax matrices with the weights as in the theta function series definition. By construction the L-matrix satisfies the Manakov triple representation instead of the Lax equation. Finally, we discuss the factorized structure of the L-matrix.

arXiv:2010.08077•SciPost Physics•Mar 4, 2021•9 citations
Fukai and Yamada introduced a Lax operator with a three-dimensional auxiliary space for the spin-1/2 XYZ chain whose entries are algebraic rather than elliptic. We give an explicit algebraic 9 × 9 intertwiner for this operator. The matrix has the 21-vertex pattern and is written directly in terms of the two spectral parameters and the three XYZ couplings. Direct reduction in the radical function field proves the local RLL relation. We also prove an exchange-unitarity identity, generic uniqueness up to scalar, and regularity after a scalar renormalization. Its invariant blocks give a factorization of the characteristic polynomial and show that, on the common finite chart, the rank-loss locus is exactly 1 − R² = 0. We make the constant gauge to the fused eight-vertex matrix explicit and obtain regular XXZ and scaled rational XXX limits. The result supplies the local intertwiner left implicit by the fusion argument; it does not constitute a new integrable model.
We describe classical top-like integrable systems arising from the quantum exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed in terms of the quantum non-dynamical R-matrix even at the classical level, where the Planck constant plays the role of the relativistic deformation parameter in the sense of Ruijsenaars and Schneider (RS). The integrable systems (relativistic tops) are described as multidimensional Euler tops, and the inertia tensors are written in terms of the quantum and classical R-matrices. A particular case of gl N system is gauge equivalent to the N-particle RS model while a generic top is related to the spin generalization of the RS model. The simple relation between quantum R-matrices and classical Lax operators is exploited in two ways. In the elliptic case we use the Belavin’s quantum R-matrix to describe the relativistic classical tops. Also by the passage to the noncommutative torus we study the large N limit corresponding to the relativistic version of the nonlocal 2d elliptic hydrodynamics. Conversely, in the rational case we obtain a new gl N quantum rational non-dynamical R-matrix via the relativistic top, which we get in a different way — using the factorized form of the RS Lax operator and the classical Symplectic Hecke (gauge) transformation. In particular case of gl2 the quantum rational R-matrix is 11-vertex. It was previously found by Cherednik. At last, we describe the integrable spin chains and Gaudin models related to the obtained R-matrix.
Exploring a mapping among n -state spin and vertex models on the square lattice, we argue that a given integrable spin model with edge weights satisfying the rapidity difference property can be formulated in the framework of an equivalent solvable vertex model. The Lax operator and the R-matrix associated to the vertex model are built in terms of the edge weights of the spin model and these operators are shown to satisfy the Yang-Baxter algebra. The unitarity of the R-matrix follows from an assumption that the vertical edge weights of the spin model satisfy certain local identities known as inversion relation. We apply this embedding to the scalar n -state Potts model and we argue that the corresponding R-matrix can be written in terms of the underlying Temperley-Lieb operators. We also consider our construction for the integrable Ashkin-Teller model and the respective R-matrix is expressed in terms of sixteen distinct weights parametrized by theta functions. We comment on the possible extension of our results to spin models whose edge weights are not expressible in terms of the difference of spectral parameters.
The construction of auxiliary matrices for the six-vertex model at a root of unity is investigated from a quantum group theoretic point of view. Employing the concept of intertwiners associated with the quantum loop algebra U q ( 2 ) at q N = 1, a three-parameter family of auxiliary matrices is constructed. The elements of this family satisfy a functional relation with the transfer matrix allowing one to solve the eigenvalue problem of the model and to derive the Bethe ansatz equations. This functional relation is obtained from the decomposition of a tensor product of evaluation representations and involves auxiliary matrices with different parameters. Because of this dependence on additional parameters, the auxiliary matrices break in general the finite symmetries of the six-vertex model, such as spin-reversal or spin-conservation. More importantly, they also lift the extra degeneracies of the transfer matrix due to the loop symmetry present at rational coupling values. The extra parameters in the auxiliary matrices are shown to be directly related to the elements in the enlarged centre Z of the algebra U q ( 2 ) at q N = 1. This connection provides a geometric interpretation of the enhanced symmetry of the six-vertex model at rational coupling. The parameters labelling the auxiliary matrices can be interpreted as coordinates on a hypersurface Spec Z ⊂ ℂ 4 which remains invariant under the action of an infinite-dimensional group G of analytic transformations, called the quantum coadjoint action.
We discuss an algebraic method for constructing eigenvectors of the transfer matrix of the eight-vertex model at the discrete coupling parameters. We consider the algebraic Bethe ansatz of the elliptic quantum group E τ,η ( sl 2 ) for the case where the parameter η satisfies 2 N η = m 1 + m 2 τ for arbitrary integers N , m 1 and m 2 . When m 1 or m 2 is odd, the eigenvectors thus obtained have not been discussed previously. Furthermore, we construct a family of degenerate eigenvectors of the XYZ spin chain, some of which are shown to be related to the sl 2 loop algebra symmetry of the XXZ spin chain. We show that the dimension of some degenerate eigenspace of the XYZ spin chain on L sites is given by N 2 L/N , if L/N is an even integer. The construction of eigenvectors of the transfer matrices of some related interaction round a face models is also discussed.
Abstract Sutherland showed that the XYZ quantum spin-chain Hamiltonian commutes with the eight-vertex model transfer matrix, so that Baxter’s subsequent tour de force proves the integrability of both. The proof requires parametrising the Boltzmann weights using elliptic theta functions and showing they satisfy the Yang-Baxter equation. We here give a simpler derivation of the integrability of the XYZ chain by explicitly constructing an extensive sequence of conserved charges from a matrix-product operator. We show that they commute with the XYZ Hamiltonian with periodic boundary conditions or an arbitrary boundary magnetic field. A straightforward generalisation yields impurity interactions that preserve the integrability. Placing such an impurity at the edge gives an integrable generalisation of the Kondo problem with a gapped bulk. We make contact with the traditional approach by relating our matrix-product operator to products of the eight-vertex model transfer matrix.
Using the intertwining matrix of the IRF-Vertex correspondence we propose a determinant representation for the generating function of the commuting Hamiltonians of the double elliptic integrable system. More precisely, it is a ratio of the normally ordered determinants, which turns into a single determinant in the classical case. With its help we reproduce the recently suggested expression for the eigenvalues of the Hamiltonians for the dual to elliptic Ruijsenaars model. Next, we study the classical counterpart of our construction, which gives expression for the spectral curve and the corresponding L-matrix. This matrix is obtained explicitly as a weighted average of the Ruijsenaars and/or Sklyanin type Lax matrices with the weights as in the theta function series definition. By construction the L-matrix satisfies the Manakov triple representation instead of the Lax equation. Finally, we discuss the factorized structure of the L-matrix.