Abstract
We construct the 1D parafermionic model based on the solution of Yang-Baxter equation and express the model by three types of fermions. It is shown that the parafermionic chain possesses both triple degenerate ground states and non-trivial topological winding number. Hence, the parafermionic model is a direct generalization of 1D Kitaev model. Both the and 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 based on Yang-Baxter equation. Different from the Majorana doubling, the holds triple degeneracy at each of energy levels. The triple degeneracy is protected by two symmetry operators of the system, ω-parity P and emergent parafermionic operator Γ, which are the generalizations of parity PM and emergent Majorana operator in Lee-Wilczek model, respectively. Both the parafermionic model and 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.
Highlights
We construct the 1D 3 parafermionic model based on the solution ofYang-Baxter equation and express the model by three types of fermions
Taking 1D p-wave Kitaev model[1] as an example, the Majorana mode appears in the topological phase where the two free Majorana fermions γ1 and γ2N can be excited without cost of energy at the two ends of the chain model and compose a non-local complex fermion, the ground state possesses double degeneracy
We have shown that both the Kitaev model and the Lee-Wilczek model can be derived from the 4 × 4 matrix representation of Yang-Baxter equation(YBE)[8]
Summary
From the view point of rational Yang-Baxterization of Temperley-Lieb algebra, when d ≤ 2, i.e. N ≤ 4, the parameter in the solution of YBE is real and the corresponding R-matrix is unitary and can be viewed as unitary evolution operator of a quantum system. There are totally non-trivial three types case, there is no cyclic permutation symmetry of parafermions on the i-th SU(3) spin site, Fi†, of r†, g† Gi† and ωFiGi (see equation (32)), but we only choose two of the three types of parafermions to represent Temperley-Lieb algebra in equation (36). It can be regarded as the extension of the algebra of Majorana doubling pointed out in ref.
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