The Application Of The Q-deformed Lie Algebra In Physics

The Yang-Baxter equation (YBE) originates in solving-function interactionmodel by C. N. Yang and statistical models by R. Baxter.Since then, investigations onquantumintegrable modelshavebeengreatlypromoted.Recently, YBE is widely appliedtoquantum computation, quantum information and quantum entanglement etc. Usually,there are three kinds of solutions for YBE: rational solution, trigonometric solution,ellipse solution. The Hamiltonian of XXZ model can be obtained through RTTrelation. XXZ model reduces to XXX model,when the parameter q=1. The accordingalgebra of trigonometric solution of YBE is quantum algebra. Sklyanin was the firstone realizing that the quantum algebra is the q-deformation of Lie algebra.In1981,Kulish and Reshetikhin also discovered the structure of quantum algebra in YBEindependently. Later on, quantum algebra is categorized abstractly to Hopf algebra byDrinfeld and Jimbo. The structure of quantum algebra adjusted by parameter q.Quantum algebra converts to Lie algebra at the moment q=1. So quantum algebra isconsidered the q-deformed Lie algebra. There are five chapters in this dissertation, thecontent as follows,Chapter one, we introduced the background knowledge of the research,itcontains theorigin and development of YBE, the relevant knowledge of Lie algebra,quantum algebra, Berry phase and topological basis.Chaptertwo, we study the XXZ chain model with periodic boundary condition.A setof topological basis realization is presented. By acting on thediferentsubspaces, we obtain the new nontrivial six-dimensional (6D)and four-dimensional Temper-Lieb matrix representations. Then it isshown that the XXZ model can be constructed from theTemperley-Liebalgebra generators. The eigenstates of XXZ model can be denotedbycombinations the set of topological bases. It is interesting that theground state is closely relatedto parameter.Chapterthree, we investigate the XXZ model’s characteristic with the twistedboundarycondition and the topological basis expression.Owing to twistboundary condition, theground state energy will change back and forthbetween and by modulating the parameter. By usingTemperley-Lieb algebra generators, the XXZ model’s Hamiltonian canbeconstructed. All the eigenstates can be expressed by topological basis,and the wholeof eigenstates’ entanglement are maximally entanglestates ().Chapterfour, we investigate the Berry phase of a spin system in a q-deformedmagnetic field. The Berry phase depends on parameter q and whichcauses the deformationof the Berry phase. When, the parameter space for the spin-half particles inmagnetic field is deformed andimmovable at the point of.ThesameHamiltonian canalso be constructed by q-deformed Lie-algebra and magneticfield（）.Chapterfive, we investigate Berry phase of quantum spin system in q-deformedmagnetic field which is definedby q-deformed Lie algebra.The corresponding general formula of q-deformed Berry phase ispresentedfor arbitrary spin system. Then we give the geometricinterpretation of q-deformed Berry phase. It is shown that the sameHamiltonian can also be generated fromWigner D-function whichsatisfiesthe Yang-Baxter equation.Finally, the conclusions and discussions are presented.