$\mathbb{Z}_3$ Parafermionic Chain Emerging From Yang-Baxter Equation

TL;DR

This paper constructs a 1D $ ext{Z}_3$ parafermionic chain from the Yang-Baxter equation, revealing topological properties and triple degeneracy, and generalizes the Majorana model to SU(3) symmetry.

Contribution

It introduces a new $ ext{Z}_3$ parafermionic model derived from Yang-Baxter solutions, expanding the understanding of topological chains beyond Majorana fermions.

Findings

The $ ext{Z}_3$ chain has triple degenerate ground states.

The model exhibits a non-trivial topological winding number.

The algebra of parafermions is demonstrated through a new 3-body Hamiltonian.

Abstract

We construct the 1D $\mathbb{Z}_3$ parafermionic model based on the solution of Yang-Baxter equation and express the model by three types of fermions. It is shown that the $\mathbb{Z}_3$ parafermionic chain possesses both triple degenerate ground states and non-trivial topological winding number. Hence, the $\mathbb{Z}_3$ parafermionic model is a direct generalization of 1D $\mathbb{Z}_2$ Kitaev model. Both the $\mathbb{Z}_2$ and $\mathbb{Z}_3$ model can be obtained from Yang-Baxter equation. On the other hand, to show the algebra of parafermionic tripling intuitively, we define a new 3-body Hamiltonian $\hat{H}_{123}$ based on Yang-Baxter equation. Different from the Majorana doubling, the $\hat{H}_{123}$ holds triple degeneracy at each of energy levels. The triple degeneracy is protected by two symmetry operators of the system, $\omega$-parity $P$($\omega=e^{{\textrm{i}\frac{2\pi}{3}}}$) and emergent parafermionic operator $\Gamma$, which are the generalizations of parity $P_{M}$ and emergent Majorana operator in Lee-Wilczek model, respectively. Both the $\mathbb{Z}_3$ parafermionic model and $\hat{H}_{123}$ can be viewed as SU(3) models in color space. In comparison with the Majorana models for SU(2), it turns out that the SU(3) models are truly the generalization of Majorana models resultant from Yang-Baxter equation.