aqora

Enlarged Symmetries of Spin Chains and Loop Models

Hubert Saleur

Aug 8, 20071 citation

Random antiferromagnetic spin chains with enlarged symmetry

Jos 'e A. Hoyos, E. Miranda

Bulletin of the American Physical SocietyMar 15, 20060 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.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/0302002Journal of Physics A Mathematical and GeneralApr 29, 200337 citations

Symmetry of quantum matrix models

C.-W. H. Lee, S. G. Rajeev

AIP conference proceedingsJan 1, 19980 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.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/0109078Journal of Physics A Mathematical and GeneralJan 21, 200233 citations

A diagrammatic approach to spin algebras

J. P. Whitehead, H. Matsumoto, Hiroomi Umezawa

Physics Letters AJul 1, 198416 citations

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