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arXiv:0712.0066

Extension of a Borel subalgebra symmetry into the sl(2) loop algebra symmetry for the twisted XXZ spin chain at roots of unity and the Onsager algebra

Tetsuo Deguchi

Published Dec 1, 2007• 2 citations
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Abstract

We discuss a conjecture that the twisted transfer matrix of the six-vertex model at roots of unity with some discrete twist angles should have the sl(2) loop algebra symmetry. As an evidence of this conjecture, we show the following mathematical result on a subalgebra of the sl(2) loop algebra, which we call a Borel subalgebra: any given finite-dimensional highest weight representation of the Borel subalgebra is extended into that of the sl(2) loop algebra, if the parameters associated with it are nonzero. Thus, if operators commuting or anti-commuting with the twisted transfer matrix of the six-vertex model at roots of unity generate the Borel subalgebra, then they also generate the sl(2) loop algebra. The result should be useful for studying the connection of the sl(2) loop algebra symmetry to the Onsager algebra symmetry of the superintegrable chiral Potts model.

DOI
10.48550/arxiv.0712.0066

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We discuss a conjecture that the twisted transfer matrix of the six-vertex model at roots of unity with some discrete twist angles should have the sl(2) loop algebra symmetry. As an evidence of this conjecture, we show the following mathematical result on a subalgebra of the sl(2) loop algebra, which we call a Borel subalgebra: any given finite-dimensional highest weight representation of the Borel subalgebra is extended into that of the sl(2) loop algebra, if the parameters associated with it are nonzero. Thus, if operators commuting or anti-commuting with the twisted transfer matrix of the six-vertex model at roots of unity generate the Borel subalgebra, then they also generate the sl(2) loop algebra. The result should be useful for studying the connection of the sl(2) loop algebra symmetry to the Onsager algebra symmetry of the superintegrable chiral Potts model.
The symmetries of the twisted XXZ spin-chain (alias the twisted six-vertex model) at roots of unity are investigated. It is shown that when the twist parameter is chosen to depend on the total spin an infinite-dimensional non-abelian symmetry algebra can be explicitly constructed for all spin sectors. This symmetry algebra is identified to be the upper or lower Borel subalgebra of the sl_2 loop algebra. The proof uses only the intertwining property of the six-vertex monodromy matrix and the familiar relations of the six-vertex Yang-Baxter algebra.
We discuss a family of operators which commute or anti-commute with the twisted transfer matrix of the six-vertex model at q being roots of unity, q 2 N = 1. The operators commute with the Hamiltonian of the XXZ spin chain under the twisted boundary conditions, and they are also valid for the inhomogeneous case. For the case of the anti-periodic boundary conditions, we show explicitly that the operators generate the sl 2 loop algebra in the sector of the total spin operator S Z ≡ N /2(mod N ). The infinite-dimensional symmetry leads to exponentially-large spectral degeneracies, as shown for the periodic boundary conditions (Deguchi T, Fabricius K and McCoy B M 2001 J. Stat. Phys. 102 701). Furthermore, we derive explicitly the sl 2 loop algebra symmetry for the periodic XXZ spin chain with an odd number of sites in the sector S Z ≡ N /2(mod N ) when q is a primitive N th root of unity with N odd. Interestingly, in the case of N = 3, various conjectures of combinatorial formulae for the XXZ spin chain with odd sites have been given by Stroganov and other authors. We also note a connection to the spectral degeneracies of the eight-vertex model.
We review the main result of cond-mat/0503564.The Hamiltonian of the XXZ spin chain and the transfer matrix of the six-vertex model has the sl 2 loop algebra symmetry if the q parameter is given by a root of unity, q 2N 0 = 1, for an integer N .We discuss the dimensions of the degenerate eigenspace generated by a regular Bethe state in some sectors, rigorously as follows: We show that every regular Bethe ansatz eigenvector in the sectors is a highest weight vector and derive the highest weight d± k , which leads to evaluation parameters a j .If the evaluation parameters are distinct, we obtain the dimensions of the highest weight representation generated by the regular Bethe state.
We examine the Onsager algebra symmetry of τ(j)τ^{(j)}τ(j)-matrices in the superintegrable chiral Potts model. The comparison of Onsager algebra symmetry of the chiral Potts model with the sl2sl_2sl2​-loop algebra symmetry of six-vertex model at roots of unity is made from the aspect of functional relations using the QQQ-operator and fusion matrices. The discussion of Bethe ansatz for both models is conducted in a uniform manner through the evaluation parameters of their symmetry algebras.
We construct the explicit Q -operator incorporated with the sl 2 -loop-algebra symmetry of the six-vertex model at roots of unity. The functional relations involving the Q -operator, the six-vertex transfer matrix and fusion matrices are derived from the Bethe equation, parallel to the Onsager-algebra-symmetry discussion in the superintegrable N -state chiral Potts model. We show that the whole set of functional equations is valid for the Q -operator. Direct calculations in certain cases are also given here for clearer illustration about the nature of the Q -operator in the symmetry study of root-of-unity six-vertex model from the functional-relation aspect.
We propose an expression for the current form of the lowering operator of the sl 2 loop algebra symmetry of the six-vertex model (XXZ spin chain) at roots of unity. This operator has poles which correspond to the evaluation parameters of representation theory which are given as the roots of the Drinfeld polynomial. We explicitly compute these polynomials in terms of the Bethe roots which characterize the highest weight states for all values of S z . From these polynomials we find that the Bethe roots satisfy sum rules for each value of S z .