Quantum Phase Transition And Entanglement Of One-dimensional Periodic Spin Models

The study of the properties of quantum spin models has been a hot topic in the field of condensed matter physics. The quantum phase transition of one dimensional spin model among them is widely studied. Traditionally, it is necessary to describe the quan-tum phase transition of the spin model with some order parameters, such as magnetic moment, magnetic susceptibility and spin correlation function. In recent years, with the development of quantum information and quantum computing, quantum phase transition has also been widely studied from the perspective of quantum information. The under-standing of quantum phase transition has been further deepened. This, in turn, promote the development of the theory and experiment in other fields, such as quantum infor-mation, biological neural networks. In addition, it is very important to study the effect of periodic structure on the phase transition of the spin model due to the periodicity of the crystal. In this thesis, quantum phase transitions of two kinds one-dimensional spin accurate model with periodic structure are solved by Jordan-Wigner transform and Bogoliubov transform, and the correlation function and quantum entanglement is used to study the phase behavior of the system.In Chapter 1, we introduce the quantum spin model and its present research situa-tion, briefly. At present, the magnetic properties, electrical properties, thermodynamic properties and dynamical properties of the models are studied deeply. The quantum phase transition at zero temperature is one of the most discussed properties. In general, it is shown that the quantum phase transition can be described by using the mean of some physical quantities. For some simple one-dimensional spin models, such as the uniform XY model, we can analytically calculate the phase transition point of the system. In recent years, some physical quantities and concepts in the field of quantum information have been used to describe the phase transition, such as quantum entanglement.In Chapter 2, We study the quantum phase transitions and quantum entanglement of one-dimensional periodic anisotropic XY models in a transverse field, whose Hamiltonian is (?). We consider period-two case, i·e. the anisotropy parameter γi takes γ and αγ alternately. It is found that the phase diagrams with the ratio a ≠-1 and a =-1 are different obviously. In the systems with α≠±1,the line (?)separates a ferromagnetic ,(FM) phase from a paramagnetic (PM) phase, while the line γ=0 is the boundary between a ferromagnetic phase along the x direction (FMx) and a ferromagnetic phase along the y direction (FMy).This is similar to these of the uniform XY chains in a transverse field(i.e. α= 1).The Ising transition and the anisotropic phase transition still exist. When α = -1,the ferromagnetic phase FMx and FMy in the uniform XY chains in a transverse field disappear and the line (?) separates a new ferromagnetic phase from a paramagnetic (PM) phase. Furthermore, the single particle energy spectrum has two zeros and the long-range correlation functions along both x and y direction are equivalent in the new phase. In order to facilitate the description, we call the gapless phase for isotropic ferromagnetic (FMxx) phase. This is because the systems has a new symmetry,i.e. after the transformationation (?),the hamiltonian is unchanged. The correlation function between the 2i -1 and 2i lattice points along x (y) direction is equal to that between the 2i and 2i -1 lattice points along y (x) direction, which leads to that the long-range correlations functions along two directions are equivalent. Finally, the relationship between quantum entanglement and phase transition of the system is studied numerically by using Von Neumann entropy. The scaling behaviour of the entanglement at every point in the isotropic ferromagnetic phase is SL～1/3 log2 L + Const,which is similar that in the anisotropic phase transition points of the uniform XY models in a transverse field.In the Chapter 3, we study the quantum phase transition by mapping the one-dimensional compass models to the XY models, whose Hamiltonian is(?) an give the phase diagram of the system according to the long-range correlation function. The models are the antiferromagnetic phase along the zdirection in the region of J1/L1 < 0, J2/L1 < 1; the ferromagnetic phase along the z direction in the region of J1/L1 > 0,J2/L1 < 1; the paramagnetic phase in the region of J2/L1 > 1 . However, the long-range correlation functions can not scale first order phase transition in J1/L1 < 0, J1/L1 > Oregion. From the type of phase transition,the system changes from the ferromagnetic phase along the z direction to the param-agnetic phase with the increase of J2/L1, which is similar to Ising phase transition. In addition, it is found that the long-range correlation functions of the system exhibits a period-four variation , which is due to the periodic structure of the system itself. At last,we study the relationship between quantum entanglement and phase transition of the system by using the Von Neumann entropy. It is found that the Von Neumann entropy has a jump in the first-order phase transition point. Compared that the concurrence, the Von Neumann entropy can not only identify the second-order phase, can clearly show the first-order phase transition. And we also find that the von Neumann entropy of the system is a fixed value without J1/L1 (except for the first phase transition point) in the case of J2/L1 and the chain length. In the vicinity of the second-order transition points,the Von Neumann entropy is still a linear relationship with the logarithm of the chain length, the coefficient is approximately equal to 1/6.