Quantum Correlation And Quantum Phase Transition Of Spin Systems

Solid-state quantum spin systems exhibit rich and unique phase-transition behaviors, which can be used to describe the novel features of many common materials and the recently discovered two-dimensional nano-materials. Therefore, they have been becoming one important research branch of condensed matter physics. These spin systems show non-local classical correlation, which exists only in quantum mechanics and is the quantum entanglement or more generally the quantum correlation. Thus they also became important physical systems to realize the quantum information processing. To study the relation between quantum correlation(or quantum entanglement) in these systems at the low or absolute zero temperature and quantum phase transition, and to examine the relation between the thermal fluctuation caused by the finite temperature and the quantum correlation, has been becoming two main directions of the research of spin system. The quantum correlation in the spin systems on two-dimensional or fractal lattices is studied in this thesis. We explored how the different structures affect the critical behaviors of the quantum correlation, and discovered the condition and mechanism for the generation long-range stable quantum correlation on the lattices with complex structures, and found the important role the space dimensionality or fractal dimensionality of the concerned system in the study of quantum correlation and quantum phase transition. The main contents of this thesis are as follows:1. The quantum entanglement of quantum Ising spin system under the transverse field on the two-dimensional square lattice at absolute zero temperature is investigated. By employing the quantum renormalization group method and concurrence as a measure, the ground-state quantum entanglement between two spins blocks is calculated. It is found that the entanglement can generate or increase when the external field increase and is close to the critical magnetic field, across this quantum critical point the magnetic field can suppress and destroy the entanglement. The entanglement between spin-blocks appears only within a certain range of the magnetic field which is around to the quantum critical point. As the size of system becomes large, the range of the magnetic field where the entanglement can exist is become small and very close to the quantum critical point. The derivative of entanglement shows a non-analytic divergent behavior, meaning that the system undergoes a typical second-order quantum phase transition. We analyze the physical mechanism how the entanglement is affected by the spin fluctuation and the correlation of fluctuation caused by the ferromagnetic-paramagnetic phase transition. According to the renormalization transformation method, the finite-size scaling behavior of entanglement is found. We find the scaling law relationship between the critical exponent of entanglement and the correlation length exponent and also discover that the system dimension plays an important role in this relationship.2. By using the quantum renormalization-group method and the concurrence as the measure of the entanglement, we study the relationship between quantum entanglement and quantum phase transition of the spin systems on the triangular lattice and Sierpiński fractal lattices, respectively. It is found that the intensity of field, the size of system and the structure of lattice can affect the ground-state entanglement between two spin blocks. On the left or right the quantum critical point, the system stays in ferromagnetic phase or paramagnetic phase, respectively. As the external field becomes large, the entanglement shows different trends: increasing or decreasing, respectively. When the size of the system becomes large, the range of the magnetic field in which the entanglement can exist becomes small and is close to the critical point. The first derivative of entanglement shows singular behavior, and the maximum or minimum value of derivative of entanglement is more and more close to the critical point. The singularity of the entanglement between spin blocks can be as the indicator to reflect the quantum phase transition. The scaling behaviors of entanglement on the different lattice structure are different(namely, with diverse critical exponents). For the triangular lattice, the space dimension determines the scaling law relationship between the entanglement critical exponent and the correlation length exponent. However, for fractal lattices, it is the fractal dimension but not the space dimension to determine this relationship.3. The quantum correlation of the quantum Heisenberg spin system on the fractal lattices at the finite temperature is studied. The quantum correlations between two end spins on one-dimensional chains, Koch curves and diamond-type hierarchical lattices are calculated by using the decimation renormalization method and the concept of quantum discord. It is found that the temperature, the anisotropy parameter, the system size and lattice structure have important influence on the quantum correlation behaviors. At finite temperature, the quantum discord shows the obvious cuspate change at the quantum critical point with the change of anisotropic parameters. For spin chains and fractal lattices, the long-range thermal quantum correlations can indicate the critical point of the quantum phase transition of the system. The regrowth of the quantum correlation with increasing temperature exhibits when the anisotropy parameter is smaller than the critical point, but the growth behavior shows when the anisotropy parameter is larger. We discussed the physical mechanism, i.e., the quantum correlation is sensitive to the energy-level crossing even at finite temperature. As the size of systems become large, the quantum correlation is more robust than entanglement, and even at unentangled states, quantum correlation can exist on fractal lattices. The effect of fractal dimension on the quantum correlation of long-range quantum is also found, i.e., the increase of fractal dimension can produce the robust long-distance quantum correlation.4. The relationship between quantum correlations and quantum phase transition on the spin glass is studied, and the effect of the random coupling is mainly discussed. We calculated the quantum discord between two non-near-neighbor spin blocks in one-dimensional random XXZ spin chain under the Dzyaloshinskii-Molriya(DM) interaction. By applying the quantum renormalization-group method and the random variables with a Gaussian distribution, we study the change trend of quantum correlation with random coupling parameters and DM interaction as the size of the system becomes large. Random coupling parameters satisfy the normal distribution and its standard deviation of reflects the disorder of the system. For the system without the random, as the DM interaction increases across the quantum critical point, the quantum correlation transfers obviously from zero to the maximum of mutations. It is found that the quantum correlation of quantum system in antiferromagnetic phase is zero, but in the corresponding quantum spin liquid phase the quantum correlation is the maximum. After introducing random disordered into the system, the spin glass effect will extend the sudden change of average quantum correlation near quantum critical point, and for this kind of random spin system, an average of the fluctuation of the quantum correlation or its standard deviation can effectively reflect the quantum phase transition of the system.