Birman-Wenzl-Murakami Algebra, Topological parameter and Berry phase

In this paper, a (3\times3)-matrix representation of the Birman-Wenzl-Murakami(BWM) algebra has been presented. Based on which, unitary matrices (A(\theta,\phi_1,\phi_2), B(\theta,\phi_1,\phi_2)) are generated via the Yang-Baxterization approach. A hamiltonian is constructed from the unitary (B(\theta,\phi)) matrix. We then study the Berry phase of the Yang-Baxter system and find the topological parameter d has relationship with berry phase.