Based On 4-body Haldane-Shastry Model Of The Yangian Algebra

One dimensional solution of long-range interaction model has very close connection between physics and mathematics.This kind of model is related to fractional quantum Hall effect,Yang-Mills theory,Yangian and many other physical problems.Haldane-Shastry(H-S)model is a typical long-range interaction model,which is a generalization of the XXX model for describing the one dimensional magnetic linkage problem.And the H-S model of spin 1/2 is a long-range interaction model,which has Yangian symmetry.However,the effect of the Yangian operator I and J have the same effect on the realization of the transition state of the quantum state.Therefore,this thesis is to construct meaningful Yangian.First of all,through the study of the three body H-S model,it is found that the transition between the eigenstates can be distinguished directly by the lie algebra,and the role of the Yangian algebra given by H-S have the same effect with lie algebras.Therefore the study continues to expand the number of particles,explore the quantum state transitions of N=4 of the H-S model.For the N=4 of H-S model energy eigenstates of the direct calculation is very complex,due to the Yang-Baxter equation and topology based on quantum entanglement,quantum computing and quantum communication has many important applications.So the H-S model to establish based on the topology,using the method of topology drawing to solve N=4 of the H-S model eigenstates.In the N=4 system the number of complete basis should have 16 bases,we can use topological base to construct a complete basis.The study found a topological base expand space is a function space of Temperley-Lieb(T-L)algebra.The H-S model of Hamiltonian group {H2,H3,H4} constructed by T-L algebraic generators,through study the spin closed chain model of H-S Hamiltonion family {H2,H3,H4} topological properties,so that the Hamiltonian of family {H2,H3,H4} of the common eigenstates can be represented by the complete basis.By using the topological drawing method can obtain the orthonormalized eigenstates,and then study the transition between the eigenstates of the H-S model,we found Yangian SUY(7)(8)(7)2(8)algebra really effective part for transitions describe is between singlet and triplet states.And Yangian algebra and Lie algebras have some same effect of describing the H-S transition.We already know that the importance of Yangian algebra is can conduct a shift operator beyond the range of Lie algebra to describe the change in different quantum states.Lie algebra operator can only change the state at the weight factors,the Yangian can be in a certain way will be connect different weight factor states together.We found this problem,so construct a new Yangian algebra realization to describe the transition of quantum states and symmetry of the N=4 of H-S model system.Finally,we constructed N=4 by adjusting the ratio of2 H,3H and4 H interactions can change the energy degeneracies of a new Hamiltonian.