ℤ3 parafermionic chain emerging from Yang-Baxter equation

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.

The ω-cyclic representation of SU(3) generators are (except T and they satisfy the following algebraic relations The braid operator can be parametrized to yield the solution of YBE by means of Yang-Baxterization [2]. Now let us recall the standard method of Yang-Baxterization. The YBE reads, and the solution isȒ Here T i is Temperley-Lieb algebra(TLA) generator, d represents the loop value of TLA and a 0 is a free parameter. If we express T i in terms of the known braid operator B i , then the solution of YBE can be obtained. In this paper, the T-L generator associated with braid operators B i can be expressed as follows, 2 ) i,i+1 .
T i satisfy T-L algebra with the loop value d = √ 3, Based on the algebraic relation in equation (1), we can verify these relations by direct calculation.
After the replacement the solutionȒ i (u) is rewritten asȒ with the angular relation When θ 1 = θ 2 = θ 3 = π 3 , the YBE turns back into the braid relation withȒ i (π/3) = ωB i . In comparison with d = √ 2 forȒ(µ) related to the Bell basis [3], µ = tan θ, and In the orthonormal basis of |r = r † |vac , |g = g † |vac and |b = b † |vac , with the fermionic condition on each site the operators u ± , s ± , d ± are expressed as Due to the constraint that the total occupation number for the fermions r, g, b is 1 on each site, In the basis of |r , |g and |b , we have the relation i.e. acting two adjoint annihilation operators on the single fermionic occupation basis. Then multiplication of the operators u ± , s ± , d ± are easily checked in the basis of |r , |g and |b , Hence Temperley-Lieb algebraic relation represented by fermions can also be checked.
The 3-body S-matrix constrained by YBE is RegardingȒ 123 as the unitary evolution of system, one can construct the 3-body Hamiltonian Here we note that due to the constraint of equation (12), there are two of the three parameters θ 1 , θ 2 and θ 3 are free Supposing that θ 1 and θ 2 is time dependent, we obtain Ignoring the constant term, we have

IV. SYMMETRY OPERATORS OFĤ123
In this section, we show that there are only two independent symmetry operators of 3-body HamiltonianĤ 123 . Let us first transform equation (19)

into matrix tensor product form under SU(3) Jordan-Wigner transformation
We have 3 .

Then the Hamiltonian readŝ
Based on the the algebraic relation in equation (1), we try to find the independent symmetry operators without linear composition. The general form of the operators can be expressed as Here a, b, x and y are to be determined, {a, b, x, y} ∈ {1, 2, 3}. The condition is that each term inĤ 123 commutes with Γ i , After direct calculation, one obtains Totally, there are 8 cases, But it is easy to check that Then only 2 independent operators are left, Making inverse SU(3) J-W transformation, we can redefine that Here we define P as the Z 3 ω-parity operator. Hence we find the two symmetry operators ofĤ 123 .