The Application Of Topological Basis On The Quantum Spin Chain Model

Research of spin chain is one of the hot research topic in recent years,the important properties of spin chain in addition to being widely used because of its itself entanglement on the physical implementation of quantum entanglement,also the main performance on the application in the field of quantum information.The Heisenberg spin chain model is the most simple spin chain model,it has a very important research value,has been used to simulate the quantum computer.Heisenberg spin chain model has a lot of kinds,this paper mainly studies the higher dimensional Heisenberg spin chain model,The higher the spin of dimension chain,the higher the Hilbert space dimension,therefore,in solving the problem of multi-body system ground state,it is a very complicated process.Topology based subspace dimension,however,are usually far less than the system of the total space dimension.Study the properties of the topology based subspace,therefore,not only can reduce the difficulty of the research questions,but also can help to study the properties of the whole spin chain model.This thesis mainly based on the theory of topology base,braid group and Temperley-Lieb algebra.We Study the related properties of spin chain through these theories.Predecessors' research is in the spin 1/2 system topology.On this basis,we study the spin 1 quantum system realization of topology.At first,this paper mainly introduces the origin of the topological base and its research status,And briefly reviews the braid group algebra,Temperley-Lieb algebra,Birman-Murakami-Wenzl algebra,those algebra and topology have a close relationship.In the spin 1/2 in the study of quantum systems,braid group and Temperley-Lieb algebra plays an important role.In the quantum system of spin 1,Birman-Murakami-Wenzl algebra used more widely.By studying the Birman-Murakami-Wenzl algebra,topology of the ground state can be found,and can study physical meaning of the topology of the ground state.Second,since Birman-Murakami-Wenzl algebra can describe the spin quantum system,so we will build the topological base associated with Birman-Murakami-Wenzl algebra,structure topology and reduce Birman-Murakami-Wenzl algebra for solving the spin chain.Finally,in theory,we will contact topology and spin chain model,We put the generator of Birman-Murakami-Wenzl linearly combined together as Hamiltonian.So we can get corresponding Hamiltonian matrixH-?_i=1^N-1?B_i+?E_i+?I,?,?,? are all real numbers,when the parameters take different values,we will Hamiltonian effect on topology based in different topological space to reduced Hamiltonian and calculate the value of the certificate and the certificate.In this way we can better study the properties of the XXZ and XXX spin chain model.