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Register a new user$mathbb{Z}_3$ Parafermionic Chain Emerging From Yang-Baxter Equation
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We construct the 1D $mathbb{Z}_3$ parafermionic model based on the solution of Yang-Baxter equation and express the model by three types of fermions. It is shown that the $mathbb{Z}_3$ parafermionic chain possesses both triple degenerate ground states and non-trivial topological winding number. Hence, the $mathbb{Z}_3$ parafermionic model is a direct generalization of 1D $mathbb{Z}_2$ Kitaev model. Both the $mathbb{Z}_2$ and $mathbb{Z}_3$ 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 $hat{H}_{123}$ based on Yang-Baxter equation. Different from the Majorana doubling, the $hat{H}_{123}$ holds triple degeneracy at each of energy levels. The triple degeneracy is protected by two symmetry operators of the system, $omega$-parity $P$($omega=e^{{textrm{i}frac{2pi}{3}}}$) and emergent parafermionic operator $Gamma$, which are the generalizations of parity $P_{M}$ and emergent Majorana operator in Lee-Wilczek model, respectively. Both the $mathbb{Z}_3$ parafermionic model and $hat{H}_{123}$ 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.
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The relationships between quantum entangled states and braid matrices have been well studied in recent years. However, most of the results are based on qubits. In this paper, We investigate the applications of 2-qutrit entanglement in the braiding associated with $mathbb{Z}_3$ parafermion. The 2-qutrit entangled state $|Psi(theta)rangle$, generated by acting the localized unitary solution $breve{R}(theta)$ of YBE on 2-qutrit natural basis, achieves its maximal $ell_1$-norm and maximal von Neumann entropy simultaneously at $theta=pi/3$. Meanwhile, at $theta=pi/3$, the solutions of YBE reduces braid matrices, which implies the role of $ell_1$-norm and entropy plays in determining real physical quantities. On the other hand, we give a new realization of 4-anyon topological basis by qutrit entangled states, then the $9times9$ localized braid representation in 4-qutrit tensor product space $(mathbb{C}^3)^{otimes 4}$ are reduced to Jones representation of braiding in the 4-anyon topological basis. Hence, we conclude that the entangled states are powerful tools in analysing the characteristics of braiding and $breve{R}$-matrix.
In this paper, the different operator forms of classical Yang-Baxter equation are given in the tensor expression through a unified algebraic method. It is closely related to left-symmetric algebras which play an important role in many fields in mathematics and mathematical physics. By studying the relations between left-symmetric algebras and classical Yang-Baxter equation, we can construct left-symmetric algebras from certain classical r-matrices and conversely, there is a natural classical r-matrix constructed from a left-symmetric algebra which corresponds to a parakahler structure in geometry. Moreover, the former in a special case gives an algebraic interpretation of the ``left-symmetry as a Lie bracket ``left-twisted by a classical r-matrix.
In this paper, several proposals of optically simulating Yang-Baxter equations have been presented. Motivated by the recent development of anyon theory, we apply Temperley-Lieb algebra as a bridge to recast four-dimentional Yang-Baxter equation into its two-dimensional counterpart. In accordance with both representations, we find the corresponding linear-optical simulations, based on the highly efficient optical elements. Both the freedom degrees of photon polarization and location are utilized as the qubit basis, in which the unitary Yang-Baxter matrices are decomposed into combination of actions of basic optical elements.
We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parakahler Lie algebra or a phase space of a Lie algebra in mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our study leads to what we call $S$-equation, which is an analogue of the classical Yang-Baxter equation. In a certain sense, the $S$-equation associated to a left-symmetric algebra reveals the left-symmetry of the products. We show that a symmetric solution of the $S$-equation gives a parakahler Lie algebra. We also show that such a solution corresponds to the symmetric part of a certain operator called ${cal O}$-operator, whereas a skew-symmetric solution of the classical Yang-Baxter equation corresponds to the skew-symmetric part of an ${cal O}$-operator. Thus a method to construct symmetric solutions of the $S$-equation (hence parakahler Lie algebras) from ${cal O}$-operators is provided. Moreover, by comparing left-symmetric bialgebras and Lie bialgebras, we observe that there is a clear analogue between them and, in particular, parakahler Lie groups correspond to Poisson-Lie groups in this sense.
Starting from the Kauffman-Lomonaco braiding matrix transforming the natural basis to Bell states, the spectral parameter describing the entanglement is introduced through Yang-Baxterization. It gives rise to a new type of solutions for Yang-Baxter equation, called the type-II that differs from the familiar solution called type-I of YBE associated with the usual chain models. The Majorana fermionic version of type-II yields the Kitaev Hamiltonian. The introduced $ell_1$ -norm leads to the maximum of the entanglement by taking the extreme value and shows that it is related to the Wigners D-function. Based on the Yang-Baxter equation the 3-body S-Matrix for type-II is explicitly given. Different from the type-I solution, the type-II solution of YBE should be considered in describing quantum information. The idea is further extended to $mathbb{Z}_3$ parafermion model based on $SU(3)$ principal representation. The type-II is in difference from the familiar type-I in many respects. For example, the quantities corresponding to velocity in the chain models obey the Lorentzian additivity $frac{u+v}{1+uv}$ rather than Galilean rule $(u+v)$. Most possibly, for the type-II solutions of YBE there may not exist RTT relation. Further more, for $mathbb{Z}_3$ parafermion model we only need the rational Yang-Baxterization, which seems like trigonometric. Similar discussions are also made in terms of generalized Yang-Baxter equation with three spin spaces ${1,frac{1}{2},frac{1}{2}}$.
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