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arXiv:0710.2764

On the Equivalent Theory of the Generalized τ^{(2)}-model and the Chiral Potts Model with two Alternating Vertical Rapidities

Shi-shyr Roan

Published Oct 15, 2007• 0 citations
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Abstract

By the Baxter's $Q_{72}$-operator method, we demonstrate the equivalent theory between the generalized $τ^{(2)}$-model (other than two special cases with a pseudovacuum state) and the $N$-state chiral Potts model with two alternating vertical rapidities, where the degenerate models are included. As a consequence, the theory of the XXZ chain model associated to cyclic representations (with the parameter $ς$) of $U_{\sf q}(sl_2)$ with ${\sf q}^N=1$ for odd $N$ is identified with either (for $ς^N=1$) the chiral Potts model with two superintegrable vertical rapidities, or (for $ς^N \neq 1$) the degenerate model for the selfdual solution of the star-triangle relation. In all these identifications, the transfer matrices $T, \hat{T}$ of the chiral Potts model (including the degenerate ones) serve as the $Q_R, Q_L$-operators of the corresponding $τ^{(2)}$-model, so that the functional relations hold as in the solvable $N$-state chiral Potts model.

DOI
10.48550/arxiv.0710.2764

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By the Baxter's Q72Q_{72}Q72​-operator method, we demonstrate the equivalent theory between the generalized τ(2)τ^{(2)}τ(2)-model (other than two special cases with a pseudovacuum state) and the NNN-state chiral Potts model with two alternating vertical rapidities, where the degenerate models are included. As a consequence, the theory of the XXZ chain model associated to cyclic representations (with the parameter ςςς) of Uq(sl2)U_{\sf q}(sl_2)Uq​(sl2​) with qN=1{\sf q}^N=1qN=1 for odd NNN is identified with either (for ςN=1ς^N=1ςN=1) the chiral Potts model with two superintegrable vertical rapidities, or (for ςN≠1ς^N \neq 1ςN=1) the degenerate model for the selfdual solution of the star-triangle relation. In all these identifications, the transfer matrices T,T^T, \hat{T}T,T^ of the chiral Potts model (including the degenerate ones) serve as the QR,QLQ_R, Q_LQR​,QL​-operators of the corresponding τ(2)τ^{(2)}τ(2)-model, so that the functional relations hold as in the solvable NNN-state chiral Potts model.
We demonstrate that the transfer matrix of the inhomogeneous N -state chiral Potts model with two vertical superintegrable rapidities serves as the Q operator of the XXZ chain model for a cyclic representation of with N th root-of-unity and representation parameter for odd N . The symmetry problem of the XXZ chain with a general cyclic representation is mapped onto the problem of studying the Q operator of some special one-parameter family of generalized τ (2) models. In particular, the spin-( N −1)/2 XXZ chain model with and the homogeneous N -state chiral Potts model at a specific superintegrable point are unified as one physical theory. By Baxter's method, developed for producing a Q 72 operator of the root-of-unity eight-vertex model, we construct the Q R , Q L and Q operators of a superintegrable τ (2) model, then identify them with transfer matrices of the N -state chiral Potts model for a positive integer N . We thus obtain a new method of producing the superintegrable N -state chiral Potts transfer matrix from the τ (2) model by constructing its Q operator.
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.
Quantum Group Theory in τ(2)τ^{(2)}τ(2)-model, Duality of τ(2)τ^{(2)}τ(2)-model and XXZ-model with Cyclic Uq(sl2){\bf U_q(sl_2)}Uq​(sl2​)-representation for qn=1{\bf q^n =1}qn=1, and Chiral Potts Model
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A new type of Yang-Baxter equation (YBE) for RRR-operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the ZNZ_NZN​ symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for N>2N>2N>2, except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated RRR-operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter RRR-matrix---realized as an intertwiner of cyclic representations of Uq(sl2)U_q(\mathfrak{sl}_2)Uq​(sl2​) at a root of unity---and Shastry's two-parameter RRR-operator obtained from the decorated YBE.
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.