It has been shown in earlier works that for Q = 0 and L a multiple of N , the ground state sector eigenspace of the superintegrable τ 2 ( t q ) model is highly degenerate and is generated by a quantum loop algebra . Furthermore, this loop algebra can be decomposed into r = ( N − 1) L / N simple algebras. For Q ≠ 0, we shall show here that the corresponding eigenspace of τ 2 ( t q ) is still highly degenerate, but splits into two spaces, each containing 2 r − 1 independent eigenvectors. The generators for the subalgebras, and also for the quantum loop subalgebra, are given generalizing those in the Q = 0 case. However, the Serre relations for the generators of the loop subalgebra are only proven for some states, tested on small systems and conjectured otherwise. Assuming their validity we construct the eigenvectors of the Q ≠ 0 ground state sectors for the transfer matrix of the superintegrable chiral Potts model.
In order to calculate correlation functions of the chiral Potts model, one only needs to study the eigenvectors of the superintegrable model. Here we start this study by looking for eigenvectors of the transfer matrix of the periodic τ 2 ( t q ) model which commutes with the chiral Potts transfer matrix. We show that the degeneracy of the eigenspace of τ 2 ( t q ) in the Q = 0 sector is 2 r , with r = ( N − 1) L / N when the size of the transfer matrix L is a multiple of N . We introduce chiral Potts model operators, different from the more commonly used generators of quantum group . From these we can form the generators of a loop algebra . For this algebra, we then use the roots of the Drinfeld polynomial to give new explicit expressions for the generators representing the loop algebra as the direct sum of r copies of the simple algebra .
In 1993, Baxter gave eigenvalues of the transfer matrix of the N -state superintegrable chiral Potts model with the spin-translation quantum number Q , where m Q = ⌊( NL − L − Q )/ N ⌋. In our previous paper we studied the Q = 0 ground-state sector, when the size L of the transfer matrix is chosen to be a multiple of N . It was shown that the corresponding τ 2 matrix has a degenerate eigenspace generated by the generators of r = m 0 simple algebras. These results enable us to express the transfer matrix in the subspace in terms of these generators E ± m and H m for m = 1, ..., r . Moreover, the corresponding 2 r eigenvectors of the transfer matrix are expressed in terms of rotated eigenvectors of H m .
Abstract Monodromy matrices of the model are known to satisfy a Yang–Baxter equation with a six-vertex R -matrix as the intertwiner. The commutation relations of the elements of the monodromy matrices are completely determined by this R -matrix. We show the reason why in the superintegrable case the eigenspace is degenerate, but not in the general case. We then show that the eigenspaces of special CSOS models descending from the chiral Potts model are also degenerate. The existence of an quantum loop algebra (or subalgebra) in these models is established by showing that the Serre relations hold for the generators. The highest weight polynomial (or the Drinfeld polynomial) of the representation is obtained by using the method of Baxter for the superintegrable case. As a byproduct, the eigenvalues of all such CSOS models are given explicitly.
We study the eigenvector problem in homogeneous superintegrable
N-state chiral Potts model (CPM) by the symmetry principal. Using duality symmetry and (spin-)inversion in CPM, together with Onsager-algebra symmetry and
sl2-loop-algebra symmetry of the superintegrable
τ(2)-model, we construct the complete
k′-dependent CPM-eigenvectors in the local spin basis for an arbitrary Onsager sector. In this paper, we present the complete classification of quantum numbers of superintegrable
τ(2)-model. Accordingly, there are four types of sectors. The relationships among Onsager sectors under duality and inversion, together with their Bethe roots and CPM-eigenvectors, are explicitly found. Using algebraic-Bethe-ansatz techniques and duality of CPM, we construct the Bethe states and the Fabricius-McCoy currents of the superintegrable
τ(2)-model through its equivalent spin-
2N−1-XXZ chain. The
τ(2)-eigenvectors in a sector are derived from the Bethe state and the
sl2-product structure determined by the Fabricius-McCoy current of the sector. From those
τ(2)-eigenvectors, the
k′-dependence of CPM state vectors in local-spin-basis form is obtained by the Onsager-algebra symmetry of the superintegrable chiral Potts quantum chain.
We demonstrate that the τ ( j ) -matrices in the superintegrable chiral Potts model possess the Onsager algebra symmetry for their degenerate eigenvalues. The Fabricius–McCoy comparison of functional relations of the eight-vertex model for roots of unity and the superintegrable chiral Potts model has been carefully analysed by identifying equivalent terms in the corresponding equations, by which we extract the conjectured relation of Q -operators and all fusion matrices in the eight-vertex model corresponding to the -relation in the chiral Potts model.
We identify the quantum group ${\Large\textsl{U}}_\textsl{w}(sl_2)$ in the
L-operator of
τ(2)-model for a generic $\textsl{w}$ as a subalgebra of
Uq(sl2) with $\textsl{w} = {\sf q}^{-2}$. In the roots of unity case, ${\sf q}=q, \textsl{w} = ω$ with
qn=ωN=1, the eigenvalues and eigenvectors of XXZ-model with the
Uq(sl2)-cyclic representation are determined by the
τ(2)-model with the induced ${\Large\textsl{U}}_ω(sl_2)$-cyclic representation, which is decomposed as a finite sum of
τ(2)-models in non-superintegrable inhomogeneous
N-state chiral Potts model. Through the theory of chiral Potts model, the
Q-operator of XXZ-model can be identified with the related chiral Potts transfer matrices, with special features appeared in the
n=2N, e.g.
N even, case. We also establish the duality of
τ(2)-models related to cyclic representations of
Uq(sl2), analogous to the
τ(2)-duality in chiral Potts model; and identify the model dual to the XXZ model with
Uq(sl2)-cyclic representation.