The Application Study Of Temperley-Lieb Algebra In The Open Heisenberg XXZ Models

In 1971,H.N.V.Temperley and E.H.Lieb proposed Temperley-Lieb algebra in jointly.Temperley-Lieb algebras grew out of a study of solvable lattice models in two-dimensional Statistical Mechanics and is related to link and knot invariants.With the research going on,Jones defined the Jones’ link and knot invariants when he research the time factor,he defined the trace function of Markov on the Temperley-Lieb algebra.We can take Temperley-Lieb quotient algebra as a Hecke algebra.A recent study shows that Temperley-Lieb algebra plays an important role in the field of quantum information.For example,in two qubit systems,the braid algebra generated by Temperley-Lieb algebra can be used to describe an important group of quantum states-Bell state in the quantum information field.In addition,Temperley-Lieb algebra also has the important value in the spin chain model,we can have the aid of Temperley-Lieb algebra to construct topology based,we can research the solution of the model and the properties of the spin chain.In addition,Temperley-Lieb algebra also has an important role in the topological quantum computing,we can use Temperley-Lieb algebra to describe the statistical behavior of quasiparticle.By studying Temperley-Lieb algebra matrix representation,this paper constructs the corresponding topological theory and with the help of topological base theory to study the related properties of Heisenberg spin chain model.The recent research shows that a kind of spin chain is closed with Temperley-Lieb algebra generator,for example,Heisenberg spin XXX model,we can establish contact between spin chain model and Temperley-Lieb algebra generator,then we research the role of the space of Temperley-Lieb algebra,it is called topological spaceTemperley-Lieb algebraic function space,and then we can study the expression of Temperley-Lieb algebra and Hamiltonian in the topological space.In this paper,we use with Temperley-Lieb algebra generator with the parameter q to construct the open boundary conditions of Heisenberg XXZ spin chain model.Thus we have researched the relationship between Temperley-Lieb algebra and Heisenberg spin chain model,finally have studied the relationship between Temperley-Lieb algebra generator and Hamiltonian of XXZ model and with the help of topological basic theory,we have studied the spin XXZ model with open boundary conditions.Such a model can be seen as the spin XXX model popularizing with the symmetry of quantum algebra U_q(su(2)).By the study,we know topological space is actually the singlet state space of the deformation of the q.In this paper,we take spin XXZ model of the four particles as an example,with the help of topological matrix theory to study the solution of the model and further study of the relationship between the ground state of the model and topological basic.