The Application Of Birman-Wenzl-Murakami Algebra And Topology In Physics

The Yang-Baxter equation originated in solving the one-dimensional±-interacting models and the statistical models on lattices which has been coined byFaddeev in the late1970s to denote a principle of integrability, i.e. exact solvability, ina wide variety of fields in physics and mathematics. Since then it has become acommon name for several classes of local equivalence transformations in statisticalmechanics, quantum field theory, differential equations, knot theory, quantum groups,and other disciplines。Braid group representations can be obtained from Yang-Baxterequation by giving a particular spectral parameter. Braid group representations of twoand three eigenvalues have direct relationship with Temperley–Lieb algebra andBirman–Wenzl–Murakami algebra respectively. Temperley–Lieb algebra andBirman–Wenzl–Murakami algebra have been widely used to construct the solutions ofYang-Baxter equation. Temperley–Lieb algebra is related to knot theory, topologicalquantum field theory, statistical physics, quantum teleportation, entangle swapping,and universal quantum computation. In quantum physics, Temperley–Lieb algebraand Birman–Wenzl–Murakami algebra natural related to spin-12and spin-1quantumsystem, respectively. To the best of our knowledge, few studies have reportedBirman–Wenzl–Murakami algebra in quantum physics. The motivation of this paperis twofold: one is that we study Birman–Wenzl–Murakami algebra and find itstopological basis states, the other is to study the physical meaning of topological basisstates. The dissertation consists of six chapters, Introduction as follows：In Chapter1, we introduction the background and the importance of our study,the general situation of quantification of Yang-Baxter equation, Braid group,Temperley–Lieb algebra, Birman–Wenzl–Murakami algebra, entanglement and Berryphase.In Chapter2, we use entangled states representations of Temperley–Lieb algebra, then we present a family of9×9-matrixrepresentations of BWMA. Finally, we study the entangled states. Onedemonstrates that the state become maximally entangled states of two qutrits In Chapter3, we obtain three topological basis states of BWMA, and we recastnine-dimensional BWMA into its three-dimensional counterpart. It is worth notingthat the topological basis states are singlet states,In Chapter4, we obtain unitary matrices A(μ;’1;’2) and B(μ;’1;’2) viaYang–Baxterization approach. Based on the solution, a Hamiltonianof the Yang–Baxter system is constructed, finally westudy the Berry phase of this system. It is interesting that in our paper, the Berryphases not only depends on the spectral parameter θ, but also depends on thetopological parameter d.In Chapter5, we study quantum spin chains of Birman-Wenzl-Murakami type.We find the Bilinear-Biquadratic spin-1Heisenberg chain, valence bond solid (VBS)model, spin-1XXX model, q-deformations of Bilinear-Biquadratic spin-1Heisenbergchain can be constructed from the Birman-Wenzl-Murakami algebra generator.In Chapter6, we study spin-1/2twist XXX model, and present a graphic methodof constructing the exact solutions for a closed four-qubit twist XXX spin chain.Finally, the conclusions and discussions are presented.