More about the doubling degeneracy operators associated with Majorana fermions and 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 $N$.