Study On Quantum Entanglement And Phasetransition Of XY Model With Transversemagnetic Field

Quantum entanglement is one of the most important concepts in quantum mechanics. Itshows perfectly nonlocality which is a unique nature of quantum mechanics. In the field ofinformation science, quantum entanglement has been widely used in areas such as quantumcomputing and quantum communication. In recent years, with the deepening of the research onquantum entanglement in spin systems, it is found that quantum entanglement plays an importantrole in quantum phase transitions. Researchers can investigate the quantum phase transitionbehavior at the quantum critical point by quantum entanglement in spin system. In this thesis,Quantum entanglement and phase transition of XY model with a transverse magnetic field arestudied by using quantum renormalization group method. The main content is as follows:By the means of site-block renormalization group method to transform Hamilton of thesystem, based on the relations of parameters before and after the transformation, the unstablefixed point (critical point) and two stable fixed ones (corresponding to two different phases ofthe spin system respectively) can be gotten. The thesis adopts the concurrence to measure theentanglement which can also be seen as order parameter to describe the critical property of thespin system. Choosing three-site as blocks, the entanglement analytical expression of twoarbitrary sites can be obtained according to the reduced density matrix of the ground state, whichis relevant to the functions of exchange coupling constant, external magnetic field and anisotropyparameter. It is easy to get the entanglement of bigger blocks as long as every parameter isconducted multistep iterations. The thesis discusses the tendency near the critical point andanalyzes the non-analytical and scaling behaviors of the entanglement. When the externalmagnetic field is fixed, the thesis studies the relations between the renormalization entanglementand every anisotropy parameter in the antiferromagnetic case and finds that the entanglementmutates at the critical point after the multistep iterations. When the entanglement is not zero,there exists the quantum correlation corresponding to the spin liquid phase. When theentanglement is zero, there only exists the classical correlation corresponding to Ising-like phase.Therefore, it shows that quantum transition takes place in the system at the critical point.Furthermore, the thesis finds that when the external magnetic field exists, the entanglement ofthe system at the critical point gradually increases as the times of iteration do until it reaches themaximum (when the entanglement is one). When every anisotropy parameter is fixed, the thesiscalculates and analyzes the variation regularity of different iteration entanglements following theexternal magnetic field in the ferromagnetic and antiferromagnetic cases. The thesis finds that inthe antiferromagnetic case, renormalization entanglement increases as the intensity of the external field heightens, while it shows oppositely in the ferromagnetic case. In order to explorethe critical property of the system, the thesis respectively calculates entanglements related toevery single anisotropy parameter and first partial derivative in the external field. It finds that theentanglement shows nonanalytic behavior near the critical point. In addition, the relationbetween the maximum of partial derivative and site numbers becomes linear, which reveals thescaling behavior of the system.