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arXiv:2508.01672 · Journal of Physics A Mathematical and Theoretical

The planar parafermion algebra: the Z N clock model and the coupled Temperley–Lieb algebra

Remy Adderton, Murray T. Batchelor

Published Jan 12, 2026• 0 citations
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Abstract

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.

DOI
10.1088/1751-8121/ae36d5

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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.
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 discuss generalizations of the Temperley–Lieb algebra in the Potts and XXZ models. These can be used to describe the addition of integrable boundary terms of different types. We use the Temperley–Lieb algebra and its one-boundary, two-boundary, and periodic extensions to classify different integrable boundary terms in the two-, three-, and four-state Potts models. The representations always lie at critical points where the algebras becomes non-semisimple and possess indecomposable representations. In the one-boundary case we show how to use representation theory to extract the Potts spectrum from an XXZ model with particular boundary terms and hence obtain the finite size scaling of the Potts models. In the two-boundary case we find that the Potts spectrum can be obtained by combining several XXZ models with different boundary terms. As in the Temperley–Lieb case, there is a direct correspondence between representations of the lattice algebra and those in the continuum conformal field theory.
The spectrum of the quantum Ising chain can be found by expressing the spins in terms of free fermions. An analogous transformation exists for clock chains with symmetry, but is of less use because the resulting parafermionic operators remain interacting. Nonetheless, Baxter showed that a certain non-Hermitian (but PT -symmetric) clock Hamiltonian is 'free', in the sense that the entire spectrum is found in terms of independent energy levels, with the striking feature that there are n possibilities for occupying each level. Here I show this directly explicitly finding shift operators obeying a generalization of the Clifford algebra. I also find higher Hamiltonians that commute with Baxter's and prove their spectrum comes from the same set of energy levels. This thus provides an explicit notion of a 'free parafermion'. A byproduct is an elegant method for the solution of the Ising/Kitaev chain with spatially varying couplings.
We investigate the Temperley–Lieb–Jones algebraic structures in isomorphic higher spin chain models with nearest neighbour interactions and symmetry. The Temperley–Lieb–Jones algebraic constructions for such chains of spins from 1 to are presented. For the case of spin-1 we also give the Temperley–Lieb–Jones algebraic representations related to the anisomorphic spin chain with -deformed symmetry. Their related spin chain Hamiltonians with and symmetries as well as the corresponding solutions of Yang–Baxter equation are also discussed.
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.