Research Of Topological Basis Realization Associated With Q Deformed Spin Chains

Since the Yang-Baxter equation was proposed,the quantum integrable spin model has received more and more attention.Spin chains are widely used in the fields of quantum entanglement,quantum computing,quantum information,etc.due to the interactions between their own particles.As a basic model,the Heisenberg spin chain model has important applications in the fields of quantum entanglement control,ferromagnetic phase transition,and ferromagnetic resonance.A generalization of the one-dimensional magnetic chain Heisenberg model,the Haldane-Shastry model,is related to physical problems such as Yang-Mills theory,Yangian quantum group theory,and fractional quantum Hall effect.Topological basis theory,as a mathematical framework,can be applied to spin-chain problems.The dimension of the topological subspace is much smaller than the dimension of the total space of the spin chain system.The properties of the topological sub space can reflect the nature of the total space to a certain extent.Previous studies have shown that the Hamiltonian of a spin chain can be constructed from Temperley-Lieb(TL)algebra and Birman-Murakami-Wenzl(BMW)algebra.The topological sub space is the working space of the TL algebra and BMW algebra generators.Based on this point of view,it is possible to study the solution of spin chains in topological space.In this thesis,the BMW algebra and TL algebra with q parameters are introduced to establish the relationships between the topological basis and the q deformed spin chains model.On the basis of previous studies on Heisenberg spin chains,a study of the Haldane-Shastry model describing long-range interactions has been added.The results show that the topological subspace is a q deformed spin singlet subspace,and all the spin singlet states falls on the topological ground states.Subsequently,the topological basis implementation of the q deformed spin chain was studied.The results show that both the non-Hermitian Heisenberg XXZ model and the q deformed Haldane-Shastry model have the quantum group Uq(sl(2))symmetry.For the spin-1 case,the non-Hermitian Heisenberg XXZ model Hamiltonian can be linearly represented by BMW algebra,and its topological space is composed of topological basis related to BMW algebra.For the spin-1/2 case,the Hamiltonian of the q deformed Haldane-Shastry model can be linearly represented by TL algebra,and its topological space is composed of topological basis related to TL algebra.By applying the Hamiltonian of the Heisenberg XXZ model and the q deformed Haldane-Shastry model to the corresponding topological basis,the Hamiltonian can be reduced,and the difficult problem of solving the high-dimensional spin chain can be solved.When q=1,the Heisenberg XXZ model is degraded to the Heisenberg XXX model,and the q deformed Haldane-Shastry model is degraded to the Haldane-Shastry model.