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arXiv:1806.03359

Integrable Chiral Potts Model and the Odd-Even Problem in Quantum Groups at Roots of Unity

Helen Au-Yang, Jacques H. H. Perk

Published Jun 8, 2018• 0 citations
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Abstract

At roots of unity the $N$-state integrable chiral Potts model and the six-vertex model descend from each other with the $τ_2$ model as the intermediate. We shall discuss how different gauge choices in the six-vertex model lead to two different quantum group constructions with different $q$-Pochhammer symbols, one construction only working well for $N$ odd, the other equally well for all $N$. We also address the generalization based on the sl$(m,n)$ vertex model.

DOI
10.48550/arxiv.1806.03359

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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.
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 identify the precise relationship between the five-parameter τ (2) -family in the N -state chiral Potts model and XXZ chains with U q ( sl 2 )-cyclic representation. By studying the Yang–Baxter relation of the six-vertex model, we discover a one-parameter family of L -operators in terms of the quantum group U q ( sl 2 ). When N is odd, the N -state τ (2) -model can be regarded as the XXZ chain of cyclic representations with . The symmetry algebra of the τ (2) -model is described by the quantum affine algebra via the canonical representation. In general, for an arbitrary N , we show that the XXZ chain with a U q ( sl 2 )-cyclic representation for q 2 N = 1 is equivalent to two copies of the same N -state τ (2) -model.
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.
Following Baxter's method of producing Q 72 -operator, we construct the Q -operator of the root-of-unity eight-vertex model for the crossing parameter with odd N where Q 72 does not exist. We use this new Q -operator to study the functional relations in the Fabricius–McCoy comparison between the root-of-unity eight-vertex model and the superintegrable N -state chiral Potts model. By the compatibility of the constructed Q -operator with the structure of Baxter's eight-vertex (solid-on-solid) SOS model, we verify the set of functional relations of the root-of-unity eight-vertex model using the explicit form of the Q -operator and fusion weights of the SOS model.