Quantum Entanglement And Quantum Phase Transition In Low-dimensional Systems

Low-dimensional quantum spin systems have been intriguing subjects in several decades. Among many achievements in this area, the phenomenon of the topological quantization of magnetization, especially the 1/3 magnetization plateau for the diamond chain and the trimerized chain, has been studied in many papers. For example, the thermodynamic properties of spin-1/2 diamond chains have been studied in many papers. The reason why these lattice models have attracted so much attention is not only that the magnetization plateaus in these models reveal interesting quantum effect, but also that they are not toy models. A magnetization plateau has been observed experimentally in Cu₃(CO₃)₂(OH)₂, which is regarded as a model substance of diamond chain, and for the spin-1/2 trimer chain compound Cu₃(P₂O₆OH)₂, a plateau has also been observed. In all these low-dimensional systems, quantum phase transitions(QPT) take a very important position. These transitions take place at zero temperature, accompanied by dramatic changes in the nature of the systems. The magnetization plateau can be explained by QPT theory. In addition, the quantum entanglement has also attracted much interest in recent years. Its non-local connotation is regarded as a valuable resource in quantum communication and information processing, and provides new perspectives for various many-body systems. In this thesis, we investigate in detail the QPTs and the quantum entanglement in the diamond chain model and the trimer chain model.For a trimerized ferromagnet(FM)-FM-antiferromagnet(AF) quantum Heisenberg chain, when the external magnetic field exceeds some threshold, the system jumps from a high entangled state to an unentangled state. The threshold is just the quantum phase transition (QPT) point for a pure trimerized chain, and it will shift away from the QPT point when a micro doping is introduced into the chain. More importantly, we find that with the variation of the intensity of doping, the threshold increases linearly, and shows good controllability. It acts as a controllable entanglement switch, which is driven by the magnetic field and its threshold is controlled by doping. It brings up an effective way to modulate the quantum entanglement, and it will be significant in quantum communication and information processing.In most cases, the nonanalytic point of the ground-state entanglement can be used to identify a quantum phase transition (QPT). While in some cases, for example, the concurrence may show a singularity in a non-critical region. In addition, in a doping model the entanglement can be influenced by the defect, thus the singularity can be changed. The reason why the singularity can derive away from the QPT point is investigated systematically, and some rules to re-build the one-to-one correspondence between QPT and the nonanalytic point of the entanglement. For example, the definition of the concurrence should be extended to minus region; for complex systems, one could use the "entanglement structure" instead of some single measure of the entanglement to detect the position of the QPT.Based on transfer-matrix density matrix renormalization group method (TMRG), a general procedure to calculate the finite-temperature pairwise entanglement of low-dimensional quantum chains is proposed. The reduced pairwise density matrix is reconstructed with TMRG, and measures of quantum entanglement can be calculated from the pairwise density matrix. The finite-temperature entanglement of the diamond chain model and the spin trimerized model, which are two typical models revealing 1/3 plateaus in the magnetization curves, is calculated. For the diamond chain model, the anisotropy coefficient A is found to have a great effect on the appearance of the magnetization plateau, and the plateau disappears whenâ–³=0.5. Moreover, our results show that the pairwise entanglement can provide information complementary to that obtained from bulk properties. For the trimerized model, the temperature dependence of the pairwise entanglement is calculated, and the threshold temperature T_c, above which the thermal entanglement vanishes, is found to be independent of the external magnetic field B. In addition, the scaling behavior of the thermal entanglement is calculated in the Trotter space. With the augmentation of the system in the Trotter direction, we find that the low-temperature entanglement shows obvious variation in the vicinity of quantum phase transition (QPT) point B_c and converges fast in non-critical regions, which provides a new way to identify QPT of 1D quantum systems.Recent years, a variant formula for DMRG is established with the help of matrix-product theory, and the new theory is used to deal with the ground-state wave-function and the time-evolution of one-dimensional systems. A matrix product system(MPS) chooses a special path in the phase diagram to undergo a quantum phase transition(QPT), and shows different behaviors compared with a traditional QPT, such as the symmetry behavior of some physical observables described in this thesis. An equation is established, which (i) helps one to understand the special behaviors of a MPS-QPTs, and (ii) can be used to detect the QPT point of a matrix product state(MPS), much simpler than usual procedures of calculating the transfer matrix or density matrix of the system. Furthermore, the discontinuity of the derivative of an observable is found to be connected directly to the turning point in the path of the MPS, but not the phase boundary point in the phase diagram, though the two are in accordance with each other in many cases.Tensor product state, or the so called projected entangled-pair state (PEPS), is the direct two-dimensional extension of MPS, and can be used to investigate two-dimensional quantum systems. We propose tensor-based symmetry equations for two-dimensional (2D) PEPSs, which can be used to construct PEPSs with predetermined symmetries. In addition, we find that one of the solutions of PEPS-symmetry-equation can be expressed as the dyadic product of one-dimensional (1D) matrix product states (MPSs), thus one can construct PEPSs through MPSs. The theory is applied to construct a spin-1 square lattice and a spin-1/2 two-layer model. The quantum entanglement of these models is calculated by the vertical density matrix algorithm, a variation of density matrix renormalization group method (DMRG). In some cases, the simple forms of the PEPSs are figured out with the help of the analysis of the quantum entanglement of the states.