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arXiv:cond-mat/0306498 · Journal of Physics A Mathematical and General

Thesl2loop algebra symmetry of the twisted transfer matrix of the six-vertex model at roots of unity

Tetsuo Deguchi

Published Dec 15, 2003• 19 citations
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Abstract

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.

DOI
10.1088/0305-4470/37/2/006

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arXiv:1508.03230•Journal of Physics A Mathematical and Theoretical•Dec 21, 2015•48 citations
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 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 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.
It has been recently discovered in the context of the six vertex or XXZ model in the fundamental representation that new symmetries arise when the anisotropy parameter (q+q−1)/2(q+q^{-1})/2(q+q−1)/2 is evaluated at roots of unity qN=1q^{N}=1qN=1. These new symmetries have been linked to an U(A1(1))U(A^{(1)}_1)U(A1(1)​) invariance of the transfer matrix and the corresponding spin-chain Hamiltonian.In this paper these results are generalized for odd primitive roots of unity to all vertex models associated with trigonometric solutions of the Yang-Baxter equation by invoking representation independent methods which only take the algebraic structure of the underlying quantum groups Uq(hatg)U_q(\\hat g)Uq​(hatg) into account. Here hatg\\hat ghatg is an arbitrary Kac-Moody algebra. Employing the notion of the boost operator it is then found that the Hamiltonian and the transfer matrix of the integrable model are invariant under the action of U(hatg)U(\\hat{g})U(hatg). For the simplest case hatg=A1(1)\\hat g=A_1^{(1)}hatg=A1(1)​ the discussion is also extended to even primitive roots of unity.
We review an algebraic method for constructing degenerate eigenvectors of the transfer matrix of the eight-vertex Cyclic Solid-on-Solid lattice model (8V CSOS model), where the degeneracy increases exponentially with respect to the system size. We consider the elliptic quantum group Eτ,η(sl2) at the discrete coupling constants: 2N η = m1 + im2τ, where N, m1 and m2 are integers. Then we show that degenerate eigenvectors of the transfer matrix of the six-vertex model at roots of unity in the sector SZ ≡ 0 ( mod N) are derived from those of the 8V CSOS model, through the trigonometric limit. They are associated with the complete N strings. From the result we see that the dimension of a given degenerate eigenspace in the sector SZ ≡ 0 ( mod N) of the six-vertex model at Nth roots of unity is given by [Formula: see text], where [Formula: see text] is the maximal value of the total spin operator SZ in the degenerate eigenspace.
Abstract We study the inhomogeneous eight-vertex model (or equivalently the XYZ Heisenberg spin-1/2 chain) with all kinds of integrable quasi-periodic boundary conditions: periodic, -twisted, -twisted or -twisted. We show that in all these cases but the periodic one with an even number of sites , the transfer matrix of the model is related, by the vertex-IRF transformation, to the transfer matrix of the dynamical six-vertex model with antiperiodic boundary conditions, which we have recently solved by means of Sklyanin’s separation of variables approach. We show moreover that, in all the twisted cases, the vertex-IRF transformation is bijective. This allows us to completely characterize, from our previous results on the antiperiodic dynamical six-vertex model, the twisted eight-vertex transfer matrix spectrum (proving that it is simple) and eigenstates. We also consider the periodic case for odd. In this case we can define two independent vertex-IRF transformations, both not bijective, and by using them we show that the eight-vertex transfer matrix spectrum is doubly degenerate, and that it can, as well as the corresponding eigenstates, also be completely characterized in terms of the spectrum and eigenstates of the dynamical six-vertex antiperiodic transfer matrix. In all these cases we can adapt to the eight-vertex case the reformulations of the dynamical six-vertex transfer matrix spectrum and eigenstates that had been obtained by T – Q functional equations, where the Q -functions are elliptic polynomials with twist-dependent quasi-periods. Such reformulations enable one to characterize the eight-vertex transfer matrix spectrum by the solutions of some Bethe-type equations, and to rewrite the corresponding eigenstates as the multiple action of some operators on a pseudo-vacuum state, in a similar way as in the algebraic Bethe ansatz framework.