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 Γ presented by Lee-Wilczek has been made. The Hamiltonian can be obtained through the new type of solution of Yang-Baxter equation, i.e. -matrix. For 2-body interaction, R(θ) gives the "superconducting" chain that is the same as 1D Kitaev chain model. The 3-body Hamiltonian commuting with Γ is derived by 3-body R₁₂₃-matrix, we thus show that the essence of the doubling degeneracy is due to [R(θ), Γ=0]. We also show that the extended Γ'-operator is an invariant of braid group BN for odd N. Moreover, with the extended Γ'-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.