Quantum Correlations In The Spin Chains And Quantum Phase Transitions

Quantum information science is an inter-discipline which appliesprinciples of the quantum mechanics to information and computer sci-ences, and is one of the most active and popular international studies. Thetasks thought to be intractable in many classical fields, such as quantumteleportation, quantum cryptography and super-dense coding, can beachieved in quantum information field. The reason why quantum infor-mation technology has an advantage over its classical counterpart is thediscovery of quantum entanglement. Quantum entanglement as a conceptthat is very different from classical mechanics, reflects the essence ofquantum mechanics: coherence, probability and special non-locality.These special properties of the entanglement, make it to be a resource ofquantum information which has been applied to the quantum communica-tion and computation. Nowadays, with the development of the quantuminformation science, spin systems show the vast potential for future ap-plications. For instance, Heisenberg model which can be realized inquantum dot systems, nuclear spin systems, electron spin systems and op-tical lattice, has been used to the studies of simulating the quantum com-puter. In addition, quantum phase transition (QPT) is an issue that is veryclosely related to Heisenberg model. A lot of works show that a singularpoint of the quantum entanglement is closely related to a transition pointof Heisenberg model. As research continues, people later found that theentanglement can’t include all the quantum correlations. Then people in-troduced quantum discord (QD) to measure the total quantum correlations.It is shown that QD perform very well in the detection of QPTs. BecauseQPTs are strongly linked to quantum correlation, it becomes an interna-tional hot topic in this aspect. This paper studies the relation between quantum correlations andQPTs in some typical spin systems. In chapter one, we briefly introducesome basic knowledge. In chapter two, we analytically investigate thenext-nearest-neighbor quantum correlations (Negativity and QD) in theXY model in thermal equilibrium with a reservoir. The result shows thatthere is no one-to-one correspondence between quantum phase transitionsand the nonanalytic property of the entanglement at zero temperature. Wealso find that when the temperature reaches to a certain point, the entan-glement completely vanishes, but there still exist non-null QD, and it canclearly detects the transition point (γ=0) of the XY model. In chapterthree, we solve the XXZ model by use of the method of accurate numeri-cal diagonalization. Through investigating the pair-wise entanglement(EoF) and geometric discord (GD) in the cyclic XXZ spin chain of eightsites in thermal equilibrium with a reservoir, we find that not only thenearest but the next-nearest GD can unambiguously spotlight the twotransition points Δ=±1, while EoF can’t do it. In chapter four, we applyquantum renormalization-group method (QRG) to study the entanglementand QPT in the one-dimensional spin-1/2Heisenberg-Ising model. Wefind the quantum phase boundary of this model, and investigate thenon-analytic and scaling behavior of von Neumann. From this scalingbehavior, we obtain the critical exponent υ=1/0.99≈1, which is con-sistent with the universality class of the Ising model. These results justifythat the QRG implementation of entanglement truly captures the criticalbehavior of this model.