The Topological Basis Realization And Application In XY Model

The quantum Heisenberg XY model is a special case of the n-vector model(the n-vector model or O(n)model is a simple system of interacting spins on a crystalline lattice.),it was intensively investigated by Lieb,Schultz,and Mattis in 1960.The classical XY model has been widely used as a model for phase transitions in materials with interacting spins.XY model exhibits the well known Kosterlitz-Thouless(KT)transition.It also can be realized in cavity QED system and in the quantum-Hall system.Meanwhile,the XY model can be used to construct the swap gate in quantum computation.Lately,the research shown that orthogonal topological basis has important applications in many aspects.With the aid of topological basis states,the four-dimensional(4D)Yang-Baxter Equation(YBE)has been reduced to the 2D YBE by nesting the Temperley-Lieb(T-L)algebra into the 4D YBE,and the result still satisfies the original operation relation.The matrix dimension can be reduced by using topological basis,which greatly reduces the complexity of the problem and turns the complex problem into a simple one.Topological basis theory also has very important applications in quantum topological computation and quantum information transmission.T-L algebraic plays an important role in quantum computation,quantum teleportation,knot theory,statistical physics and topological quantum field theory.The T-L algebra first appeared in statistical mechanics as a tool analyze various interrelated lattice models and was related to link and knot invariants.Till now there are many more models,which are based on the T-L algebra representations.In this paper,we first study the relationship between the XY model and the T-L algebra,It is shown that the Hamiltonian of the XY model can be constructed by linear combination of the generators of the T-L algebra.On this basis,we construct a new set of topological ground states,and find the rest of the orthogonal complete states.Then we study the reduced representation of the generators of T-L algebras in different subspaces.Finally,we establish the relation between the XY model and the T-L algebra,and further study the special physical properties of the topological ground state in the XY model.It is found that the topological ground state is the eigenstate of the XY model,and the energy ground state of the system falls on a topological ground state.Meanwhile,we also study the spin realization of topological basis in odd form,and discuss the reduced matrix representation of T-L algebraic generator in topological space.