Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1206.4356

Quantum Group Theory in $τ^{(2)}$-model, Duality of $τ^{(2)}$-model and XXZ-model with Cyclic ${\bf U_q(sl_2)}$-representation for ${\bf q^n =1}$, and Chiral Potts Model

Shi-shyr Roan

Published Jun 19, 2012• 1 citation
arXivHTMLPDF

Abstract

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 $U_{\sf q} (sl_2)$ with $\textsl{w} = {\sf q}^{-2}$. In the roots of unity case, ${\sf q}=q, \textsl{w} = ω$ with $q^{\bf n} = ω^N = 1$, the eigenvalues and eigenvectors of XXZ-model with the $U_q (sl_2)$-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 ${\bf n}=2N$, e.g. $N$ even, case. We also establish the duality of $τ^{(2)}$-models related to cyclic representations of $U_q (sl_2)$, analogous to the $τ^{(2)}$-duality in chiral Potts model; and identify the model dual to the XXZ model with $U_q (sl_2)$-cyclic representation.

DOI
10.48550/arxiv.1206.4356

Related papers

The transfer matrix of a superintegrable chiral Potts model as theQoperator of root-of-unity XXZ chain with cyclic representation of U_{\mathsf {q}}(sl_2)

Shi-shyr Roan

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.

arXiv:0705.2856•Journal of Statistical Mechanics Theory and Experiment•Sep 27, 2007•11 citations

On theτ(2)-model in the chiral Potts model and cyclic representation of the quantum groupUq(sl2)

Shi-shyr Roan

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.

arXiv:0806.0216•Journal of Physics A Mathematical and Theoretical•Jan 21, 2009•7 citations

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

Shi-shyr Roan

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.

arXiv:0710.2764•Oct 15, 2007•0 citations

Duality and symmetry in chiral Potts model

Shi-shyr Roan

We discover an Ising-type duality in the general N -state chiral Potts model, which is the Kramers–Wannier duality of a planar Ising model when N = 2. This duality relates the spectrum and eigenvectors of one chiral Potts model at a low temperature (of small k ') to those of another chiral Potts model at a high temperature (of k ' −1 ). The τ (2) model and chiral Potts model on the dual lattice are established alongside of the dual chiral Potts models. With the aid of this duality relation, we exact a precise relationship between the Onsager-algebra symmetry of a homogeneous superintegrable chiral Potts model and the sl 2 -loop-algebra symmetry of its associated spin- ( N −1)/2 XXZ chain through the identification of their eigenstates.

arXiv:0905.1924•Journal of Statistical Mechanics Theory and Experiment•Aug 10, 2009•2 citations

Duality and Symmetry in Chiral Potts Model

Shi-shyr Roan

We discover a Ising-type duality in the general N-state chiral Potts model, which is the Kramers-Wannier duality of planar Ising model when N = 2. This duality relates the spectrum and eigenvectors of one chiral Potts model at a low temperature (of small k ′) to those of another chiral Potts model at a high temperature (of k ′−1). The τ (2)-model and chiral Potts model on the dual lattice are established alongside of the dual chiral Potts models. With the aid of this duality relation, we exact a precise relationship between the Onsager-algebra symmetry of a homogeneous superintegrable chiral Potts model and the sl2-loop-algebra symmetry of its associated spin- N−1 2 XXZ chain through the identification of their eigenstates.

•Jan 1, 2009•5 citations

Eigenvectors in the superintegrable model I: {\frak{sl}}_2 generators

Helen Au-Yang, Jacques H. H. Perk

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 .

arXiv:0710.5257•Journal of Physics A Mathematical and Theoretical•Jun 12, 2008•24 citations
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
We identify the quantum group \textslU\textslw(sl2){\Large\textsl{U}}_\textsl{w}(sl_2)\textslU\textsl​w(sl2​) in the LLL-operator of τ(2)τ^{(2)}τ(2)-model for a generic \textslw\textsl{w}\textslw as a subalgebra of Uq(sl2)U_{\sf q} (sl_2)Uq​(sl2​) with \textslw=q−2\textsl{w} = {\sf q}^{-2}\textslw=q−2. In the roots of unity case, q=q,\textslw=ω{\sf q}=q, \textsl{w} = ωq=q,\textslw=ω with qn=ωN=1q^{\bf n} = ω^N = 1qn=ωN=1, the eigenvalues and eigenvectors of XXZ-model with the Uq(sl2)U_q (sl_2)Uq​(sl2​)-cyclic representation are determined by the τ(2)τ^{(2)}τ(2)-model with the induced \textslUω(sl2){\Large\textsl{U}}_ω(sl_2)\textslUω​(sl2​)-cyclic representation, which is decomposed as a finite sum of τ(2)τ^{(2)}τ(2)-models in non-superintegrable inhomogeneous NNN-state chiral Potts model. Through the theory of chiral Potts model, the QQQ-operator of XXZ-model can be identified with the related chiral Potts transfer matrices, with special features appeared in the n=2N{\bf n}=2Nn=2N, e.g. NNN even, case. We also establish the duality of τ(2)τ^{(2)}τ(2)-models related to cyclic representations of Uq(sl2)U_q (sl_2)Uq​(sl2​), analogous to the τ(2)τ^{(2)}τ(2)-duality in chiral Potts model; and identify the model dual to the XXZ model with Uq(sl2)U_q (sl_2)Uq​(sl2​)-cyclic representation.
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.
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 discover an Ising-type duality in the general N -state chiral Potts model, which is the Kramers–Wannier duality of a planar Ising model when N = 2. This duality relates the spectrum and eigenvectors of one chiral Potts model at a low temperature (of small k ') to those of another chiral Potts model at a high temperature (of k ' −1 ). The τ (2) model and chiral Potts model on the dual lattice are established alongside of the dual chiral Potts models. With the aid of this duality relation, we exact a precise relationship between the Onsager-algebra symmetry of a homogeneous superintegrable chiral Potts model and the sl 2 -loop-algebra symmetry of its associated spin- ( N −1)/2 XXZ chain through the identification of their eigenstates.
We discover a Ising-type duality in the general N-state chiral Potts model, which is the Kramers-Wannier duality of planar Ising model when N = 2. This duality relates the spectrum and eigenvectors of one chiral Potts model at a low temperature (of small k ′) to those of another chiral Potts model at a high temperature (of k ′−1). The τ (2)-model and chiral Potts model on the dual lattice are established alongside of the dual chiral Potts models. With the aid of this duality relation, we exact a precise relationship between the Onsager-algebra symmetry of a homogeneous superintegrable chiral Potts model and the sl2-loop-algebra symmetry of its associated spin- N−1 2 XXZ chain through the identification of their eigenstates.
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 .