Entanglement and Berry Phase in a $9\times 9$ Yang-Baxter system

A M-matrix which satisfies the Hecke algebraic relations is presented. Via the Yang-Baxterization approach, we obtain a unitary solution $\breve{R}(\theta ,{\phi }_1,{\phi }_2)$ of Yang-Baxter Equation. It is shown that any pure two-qutrit entangled states can be generated via the universal $\breve{R}$-matrix assisted by local unitary transformations. A Hamiltonian is constructed from the $\breve{R}$-matrix, and Berry phase of the Yang-Baxter system is investigated. Specifically, for ${\phi }_1={\phi }_2$, the Hamiltonian can be represented based on three sets of SU(2) operators, and three oscillator Hamiltonians can be obtained. Under this framework, the Berry phase can be interpreted.