  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 $N$.
