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arXiv:1109.6503 · Journal of Physics A Mathematical and Theoretical

New symmetries of the chiral Potts model

Jens Fjelstad, Teresia Månsson

Published Mar 27, 2012• 4 citations
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Abstract

In this paper a hithertho unknown symmetry of the three-state chiral Potts model is found consisting of two coupled Temperley-Lieb algebras. From these we can construct new superintegrable models. One realisation is in terms of a staggered isotropic XY spin chain. Further we investigate the importance of the algebra for the existence of mutually commuting charges. This leads us to a natural generalisation of the boost-operator, which generates the charges.

DOI
10.1088/1751-8113/45/15/155208

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In this paper a hithertho unknown symmetry of the three-state chiral Potts model is found consisting of two coupled Temperley-Lieb algebras. From these we can construct new superintegrable models. One realisation is in terms of a staggered isotropic XY spin chain. Further we investigate the importance of the algebra for the existence of mutually commuting charges. This leads us to a natural generalisation of the boost-operator, which generates the charges.
Abstract The Hamiltonian of the N-state superintegrable chiral Potts (SICP) model is written in terms of a coupled algebra defined by N − 1 types of Temperley–Lieb generators. This generalises a previous result for N = 3 obtained by Fjelstad and Månsson (2012 J. Phys. A: Math. Theor. 45 155208). A pictorial representation of a related coupled algebra is given for the N = 3 case which involves a generalisation of the pictorial presentation of the Temperley–Lieb algebra to include a pole around which loops can become entangled. For the two known representations of this algebra, the N = 3 SICP chain and the staggered spin-1/2 XX chain, closed (contractible) loops have weight 3 and weight 2, respectively. For both representations closed (non-contractible) loops around the pole have weight zero. The pictorial representation provides a graphical interpretation of the algebraic relations. A key ingredient in the resolution of diagrams is a crossing relation for loops encircling a pole which involves the parameter ρ = e 2πi/3 for the SICP chain and ρ = 1 for the staggered XX chain. These ρ values are derived assuming the Kauffman bracket skein relation.
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.
It is shown that there is a hidden connection between the two well-studied sequences of the Temperley-Lieb (TL) integrable models -- the qqq-state quantum Potts (QP) models at the self-dual points and the staggered SU(n){\rm SU}(n)SU(n) spin-sss chains with n=2s+1n=2s+1n=2s+1 (s≥1s \ge 1s≥1), in addition to the uniform SU(2){\rm SU}(2)SU(2) spin-1/21/21/2 Heisenberg model. For each sequence, symmetry group factorization arises, in the sense that if qqq is factorized into q1q_1q1​ and q2q_2q2​, then the qqq-state QP model is unitarily equivalent to a combined QP model with the symmetry group Sq1×Sq2{\rm S}_{q_1} \times {\rm S}_{q_2}Sq1​​×Sq2​​ or if nnn is factorized into n1n_1n1​ and n2n_2n2​, then the staggered SU(n){\rm SU}(n)SU(n) spin-sss chain with the symmetry group SU(n){\rm SU}(n)SU(n) is unitarily equivalent to a combined staggered SU(n1)×SU(n2){\rm SU}(n_1) \times {\rm SU}(n_2)SU(n1​)×SU(n2​) spin chain with the symmetry group SU(n1)×SU(n2){\rm SU}(n_1) \times {\rm SU}(n_2)SU(n1​)×SU(n2​), valid for both ferromagnetic (FM) and antiferromagnetic (AF) cases. Moreover, the FM (AF) staggered SU(n){\rm SU}(n)SU(n) spin-sss chain is unitarily equivalent to the AF (FM) qqq-state QP model with q=n2q=n^2q=n2, as long as the size of the AF (FM) staggered SU(n){\rm SU}(n)SU(n) spin-sss chain is doubled. A combination of the two distinct types of unitary equivalences yields a family of models such that they are essentially identical, but appear in different guises. Some physical implications for unitary equivalence among different TL integrable models are clarified.
We prove an exact finite-volume symmetry formula for two-point functions in the periodic NNN-state superintegrable chiral Potts spin chain. We show that, for every chain length LLL and every simultaneous eigenvector of the Hamiltonian and the one-site translation operator, the correlations satisfy ⟨Z0rZR†r⟩∗=⟨Z0rZL−R†r⟩\langle Z_0^r Z_R^{\dagger r}\rangle^*=\langle Z_0^r Z_{L-R}^{\dagger r}\rangle⟨Z0r​ZR†r​⟩∗=⟨Z0r​ZL−R†r​⟩ for 1⩽r⩽N−11\leqslant r\leqslant N-11⩽r⩽N−1. Hence, whenever LLL is even, the midpoint correlation ⟨Z0rZL/2†r⟩\langle Z_0^r Z_{L/2}^{\dagger r}\rangle⟨Z0r​ZL/2†r​⟩ is real. Then we generalise the three-state chain case to arbitrary NNN and to every translation eigensector. This resolves a conjecture of Fabricius and McCoy.
Abstract The Hamiltonian of the N -state clock model is written in terms of a coupled Temperley–Lieb (TL) algebra defined by N − 1 types of TL generators. This generalizes a previous result for N = 3 obtained by Fjelstad and Månsson (2012 J. Phys. A 45 155208). The Z N -symmetric clock chain Hamiltonian expressed in terms of the coupled TL algebra generalizes the well known correspondence between the N -state Potts model and the TL algebra. The algebra admits a pictorial description in terms of a planar algebra involving parafermionic operators attached to n strands. A key ingredient in the resolution of diagrams is the string Fourier transform. The pictorial presentation also allows a description of the Hilbert space. We also give a pictorial description of the representation related to the staggered XX spin chain. Just as the pictorial representation of the TL algebra has proven to be particularly useful in providing a visual and intuitive way to understand and manipulate algebraic expressions, it is anticipated that the pictorial representation of the coupled TL algebra may lead to further progress in understanding various aspects of the Z N clock model, including the superintegrable chiral Potts model.