The planar parafermion algebra: The $\mathbb{Z}_{N}$ clock model and the coupled Temperley-Lieb algebra

TL;DR

This work generalizes the coupled Temperley-Lieb algebra from the $N=3$ case to the full $ Z_N$ clock model by introducing a planar parafermion algebra framework. The authors define a Gerstenhaber-like coupled TL presentation $ ext{cTL}_n(\sqrt{N})$ with $N$-fold labeled generators $e_i^{(k)}$, realized diagrammatically in the planar parafermion algebra PF$_n$ and linked to parafermion operators via a Fradkin–Kadanoff transformation. A central tool is the string Fourier transform, which enforces the necessary cubic relations and provides a rotation mechanism in the diagrammatic calculus, enabling a pictorial description of both the clock model Hamiltonian and the Hilbert space. The paper also connects the cTL algebra to the staggered XX spin chain and discusses a pictorial representation that ties to chromatic algebras, suggesting potential pathways to address integrability and spectrum generation, particularly for the superintegrable chiral Potts model. Overall, the framework offers a cohesive diagrammatic and algebraic approach to $ Z_N$ clock models, with broad implications for understanding their integrable structure and Hilbert-space decomposition.

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 J. F. Fjelstad and T. Månsson [J. Phys. A {\bf 45} (2012) 155208]. The $\mathbb{Z}_{N}$-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 $\mathbb{Z}_{N}$ clock model, including the superintegrable chiral Potts model.