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Physics Letters A

Note on the Yang-Baxter equations for generalized Baxter models

Hans Christian Öttinger, Josef Honerkamp

Published Mar 1, 1982• 9 citations
Publisher
DOI
10.1016/0375-9601(82)90609-0

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arXiv:1507.05269•Jul 19, 2015•1 citation

Duality relation for a finite Baxter model

D. Merlini, C. Gruber

Physics Letters A•Sep 1, 1972•2 citations
The Yang-Baxter equation appear in various situations in physics and mathematics. For example it arises as a consistency condition in integrable models. The reflection equation (boundary Yang-Baxter equation) is a generalization of the Yang-Baxter equation to systems with a boundary. A further generalization to systems with defects which admits both reflection and transmission can be made, which results in reflection-transmission Yang-Baxter equations.In this thesis the Yang-Baxter equation and the reflection equation are presented. Representations of the Temperley-Lieb algebra and the blob algebra are used to construct matrices which solve the respective equations. For the reflection-transmission Yang-Baxter equations, steps toward a solution are taken by using a similar approach as for the first two cases, namely by finding an algebra whose representations can be used to construct matrices which solve the equations.
Z3\mathbb{Z}_3Z3​ Parafermionic Chain Emerging From Yang-Baxter Equation
We construct the 1D Z3\mathbb{Z}_3Z3​ parafermionic model based on the solution of Yang-Baxter equation and express the model by three types of fermions. It is shown that the Z3\mathbb{Z}_3Z3​ parafermionic chain possesses both triple degenerate ground states and non-trivial topological winding number. Hence, the Z3\mathbb{Z}_3Z3​ parafermionic model is a direct generalization of 1D Z2\mathbb{Z}_2Z2​ Kitaev model. Both the Z2\mathbb{Z}_2Z2​ and Z3\mathbb{Z}_3Z3​ 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 H^123\hat{H}_{123}H^123​ based on Yang-Baxter equation. Different from the Majorana doubling, the H^123\hat{H}_{123}H^123​ holds triple degeneracy at each of energy levels. The triple degeneracy is protected by two symmetry operators of the system, ωωω-parity PPP(ω=ei2π3ω=e^{{\textrm{i}\frac{2π}{3}}}ω=ei32π​) and emergent parafermionic operator ΓΓΓ, which are the generalizations of parity PMP_{M}PM​ and emergent Majorana operator in Lee-Wilczek model, respectively. Both the Z3\mathbb{Z}_3Z3​ parafermionic model and H^123\hat{H}_{123}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.