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

TL;DR

This paper introduces a coupled Temperley-Lieb algebra framework for the $N$-state clock model, providing a new pictorial and algebraic perspective that generalizes known models and offers insights into their structure.

Contribution

It generalizes the coupled TL algebra to the $N$-state clock model and introduces a pictorial planar algebra representation involving parafermionic operators.

Findings

Algebraic formulation of the $N$-state clock model using coupled TL algebra.

Pictorial planar algebra description with parafermionic operators.

Potential for new insights into the superintegrable chiral Potts model.

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. M\r{a}nsson [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.