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 state
s 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 $P$($omega=e^{{textrm{i}frac{2pi}{3}}}$) 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.
A new realization of doubling degeneracy based on emergent Majorana operator $Gamma$ presented by Lee-Wilczek has been made. The Hamiltonian can be obtained through the new type of solution of Yang-Baxter equation, i.e. $breve{R}(theta)$-matrix. For
2-body interaction, $breve{R}(theta)$ gives the superconducting chain that is the same as 1D Kitaev chain model. The 3-body Hamiltonian commuting with $Gamma$ is derived by 3-body $breve{R}_{123}$-matrix, we thus show that the essence of the doubling degeneracy is due to $[breve{R}(theta), Gamma]=0$. We also show that the extended $Gamma$-operator is an invariant of braid group $B_N$ for odd $N$. Moreover, with the extended $Gamma$-operator, we construct the high dimensional matrix representation of solution to Yang-Baxter equation and find its application in constructing $2N$-qubit Greenberger-Horne-Zeilinger state for odd $N$.
This paper investigates the physical effects of Yang-Baxter equation (YBE) to quantum entanglements through the 3-body S-matrix in entangling parameter space. The explicit form of 3-body S-matrix $breve{R}_{123}(theta,varphi)$ based on the 2-body S-m
atrices is given due to the factorization condition of YBE. The corresponding chain Hamiltonian has been obtained and diagonalized, also the Berry phase for 3-body system is given. It turns out that by choosing different spectral parameters the $breve{R}(theta,varphi)$-matrix gives GHZ and W state respectively. The extended 1-D Kitaev toy model has been derived. Examples of the role of the model in entanglement transfer are discussed.
