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arXiv:cond-mat/0602427 · Symmetry Integrability and Geometry Methods and Applications

On the Degenerate Multiplicity of the sl 2 Loop Algebra for the 6V Transfer Matrix at Roots of Unity

Tetsuo Deguchi

Published Feb 17, 2006• 1 citation
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Abstract

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.

DOI
10.3842/sigma.2006.021

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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.
The sl2sl_2sl2​ loop algebra symmetry of the XXZ spin chain: an algorithm for the degeneracy of a regular Bethe state(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )
The Hamiltonian of the XXZ spin chain has the sl2sl_{2}sl2​ loop algebra symmetry if the qqq parameter is given by a root of unity, q02N=1q_{0}^{2N}=1q02N​=1 , for an integer NNN .We review an algorithm for determining the dimensions of the degenerate eigenspace generated by a given regular Bethe state in some sectors.The proof of the algorithm has been formulated in Ref.[3] as follows: We show in some sectors that regular Bethe ansatz eigenvectors are highest weight vectors and generate irreducible representations of the sl2sl_{2}sl2​ loop algebra; We prove that every finite-dimensional highest weight representation of the sl2sl_{2}sl2​ loop algebra is irreducible; We then derive the dimensions of the highest weight representation generated by a given regular Bethe state through the Drinfeld polynomial, which is expressed explicitly in terms of the Bethe roots.
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XXZ Bethe states as highest weight vectors of the sl2sl_2sl2​ loop algebra at roots of unity
We prove some part of the conjecture that regular Bethe ansatz eigenvectors of the XXZ spin chain at roots of unity are highest weight vectors of the sl2sl_2sl2​ loop algebra. Here qqq is related to the XXZ anisotropic coupling ΔΔΔ by Δ=(q+q−1)/2Δ=(q+q^{-1})/2Δ=(q+q−1)/2, and it is given by a root of unity, q2N=1q^{2N}=1q2N=1, for a positive integer NNN. We show that regular XXZ Bethe states are annihilated by the generators xˉk+{\bar x}_k^{+}xˉk+​'s, for any NNN. We discuss, for some particular cases of N=2, that regular XXZ Bethe states are eigenvectors of the generators of the Cartan subalgebra, hˉk{\bar h}_khˉk​'s. Here the loop algebra U(L(sl2))U(L(sl_2))U(L(sl2​)) is generated by xˉk±{\bar x}_k^{\pm}xˉk±​ and hˉk{\bar h}_khˉk​ for k∈Zk \in {\bf Z}k∈Z, which are the classical analogues of the Drinfeld generators of the quantum loop algebra Uq(L(sl2))U_q(L(sl_2))Uq​(L(sl2​)). A representation of U(L(sl2))U(L(sl_2))U(L(sl2​)) is called highest weight if it is generated by a vector ΩΩΩ which is annihilated by the generators xˉk+{\bar x}_k^{+}xˉk+​'s and such that ΩΩΩ is an eigenvector of the hˉk{\bar h}_khˉk​'s. We also discuss the classical analogue of the Drinfeld polynomial which characterizes the irreducible finite-dimensional highest weight representation of U(L(sl2))U(L(sl_2))U(L(sl2​)).
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 .
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.
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.