The Entanglement Of Some 1D Spin Systems

In this thesis, we study the entanglement properties of some spin systems. Firstly we introduce some conceptions of the entanglement in quantum world, and explain three measures of entanglement, in which we emphasize the entanglement of the formation. Then we introduce the conception of concurrence defined by W. K. Woottcrs, which is related to the entanglement of formation, and with which we can study the behaviors of the entanglement. As a simple example, the entanglement of two qubits systems are studied analytically.Secondly we study the entanglement of anisotropic XY chains in a transverse magnetic field. The behaviors of the entanglement of three cases of the XY chain: uniform chain; period-two and period-three chains, are studied by using the concurrence. It is found that all the critical points obtained by the entanglement are consistent to the analytical results. For the anisotropic XY chains in a transverse field, we find that the concurrence is sensitive to the boundary condition. For finite size spin chain of even spins with the periodic boundary condition, the concurrence has a gap at the critical point, whereas it is continuous for odd case under the same boundary condition. And the gap disappears as the number of the spins goes to infinite. This is because of the boundary term which is neglected in calculating the concurrence while it is neglegible for studies of the other physical quantities. With the anti-periodic boundary condition, the gap of the concurrence for the even-spin chain will disappear, and the concurrence at the vicinity of the critical point have the same scaling law with that of the odd case with the periodic boundary condition. For the period-two and -three chains, there is more than one critical point at some parameter region. And we also find that the concurrence at the vicinity of all the critical points have the similar scaling la.w. Therefore, all these critical points belong to the same university class.Finally we study the thermal entanglement and the evolution of the entanglement for the two-spin anisotropic XY model in a transverse field. We find that for any anisotropic parameter, there exist a region of the exchange interaction, in which there are three threshold temperature. To study the evolution of the entanglement we discuss different behavior of the evolution for different initial condition.