No abstract

In this paper, we use entangled states to construct 9 × 9-matrix representations of Temperley-Lieb algebra (TLA), then a family of 9 × 9-matrix representations of Birman-Wenzl-Murakami algebra (BWMA) have been presented. Based on which, three topological basis states have been found.And we apply topological basis states to recast nine-dimensional BWMA into its three-dimensional counterpart. Finally, we find the topological basis states are spin singlet states in special case.

In this paper, it is shown that the XXX model can be constructed from the Temperley-Lieb algebra (TLA) generator. We find that the topological basis states are the two eigenstaes of a closed four-qubit Heisenberg XXX spin chain. Specifically, the spin single states and the energy single state of the system all fall on the topological basis states. It is worth mentioning that for the closed 2N -qubit (N = 2, 3, 4, . . .) Heisenberg XXX spin chain, all the topological basis states for 2N particles are the spin single states of the system. And the number of the topological basis states is equal to the number of the spin single states of the system, which is (2N )! N !(N +1)! .

We investigate the Berry phase for quantum spin system in q-deformed magnetic field. Through the q-deformed Lie algebra, we found a set of magnetic field expressions which contain the parameter q. The Berry phase of the system could be adjusted by the parameter q. The corresponding general formula of the q-deformed Berry phase is presented for an arbitrary spin system. The same Hamiltonian can also be generated from the Wigner D-function which satisfies the Yang-Baxter equation.

The quantum phase transition for the "q-deformed" Yang-baxter Hamiltonian has been discussed. The calculation shows when the deformed parameter q approaches 1, there exists a quantum critical point for spectral parameter θ . In this Yang-Baxter system, quantum entanglement and the geometric phase can characterize quantum phase transition.