Quantum Phase Transitions And Critical Phenomena For One-dimensional And Quasi-one-dimensional Many-Body Systems

By utilizing the infinite time evolving block decimation(iTEBD) and gradient method for spin ladder systems in the context of tensor network algorithm,we have studied one-dimensional and quasi-one-dimensional quantum many-body systems in the thermodynamic limit, respectively. Ground-state wave functions of spin chains and spin ladders with high accuracy are obtained by numerical approximation. Then reduced matrices are calculated by tensors consisting of the ground-state wave functions. By acting observable operators on the reduced matrices, many interesting physical observables, such as ground-state fidelity per lattice site and order parameters can be figured out.The first chapter introduces some background concepts closely related to this work. Compared with thermal phase transitions, the definition of quantum phase transitions(QPTs) is given firstly. Then ground-state energy spectrum, fidelity, spontaneous symmetry breaking(SSB), local order parameter, renormalization group theory and scaling theory are introduced. Lastly, von Neumann entropy with the area law is introduced.In Chapter two, the first part indroduces the quasi-one-dimensional gradient method in the representation of tensor network algorithm for spin ladder systems. Then approximate ground-state wave functions can be updated with this algorithm. In the second part, combining with ground-state fidelity per lattice site, the Heisenberg frustrated spin two-leg ladder with cross-coupled interaction is studied as there is a controversy about the existence of local columnar dimer phase(CD) in the phase diagram. The ground-state fidelity phase diagram detects a QPTs point and a continuous QPTs proccess because the internal structures of ground-state wave functions lead to the appearance of a pinch point and in the renormalization flow direction, the ground-state fidelity per lattice site decreases monotonically. At the same time, one could use fidelity to make a further check on the existence of the local CD phase, which has not been applied to ladder systems before. Based on the Landau theory for QPTs, if one phase could be characterized by a local order parameter, the phase is induced by a SSB and the phase should have degenerate ground-states. The result shows no degenerate ground-states, i.e., the local CD phase should not exist. Then from a perspective of order parameters including Haldane phase, rung-singlet phase and CD phase, we characterize the phases. At last, a particular symmetry of Hamiltonian posing a strong constraint on the nature of the phase diagram for this model is considered. All the evidences support the phase diagram is composed without CD phase.In Chapter three, the first part introduces about the observable-geometric entanglement(GE). Then combining with fidelity and GE, we studied another controversy about the order of QPTs of the Heisenberg frustrated spin two-leg ladder with cross-coupled interaction. To be more convincing, we offer three evidences: GE, string order parameters and ground-state fidelity per lattice site. All three evidences show a second-order QPTs in weak coupling area while a first-order QPTs in strong coupling area. From the present results we can give the conclusion above. More are to be done in our future work to detect the crossover point.In Chapter four, the first part introduces about the iTEBD algorithm. Then based on the iTEBD algorithm and other observables deduced from reduced density matrices of ground-state wave functions, we studied phase diagram of an extended quantum compass model(EQCM). It is the first time to study this model from a perspective of order parameters in depth. In order to characterize the phase properly in the phase diagram, we followed three steps originated from the way that fidelity characterizes phases. Firstly, detect the QPTs points from ground-state energies and their derivatives. Secondly, characterize the local order parameters in a given phase(if any). To obtain the local order parameters, we carefully examine two-site correlations and local magnetizations. Thirdly, characterize the non-local long-range phases. To obtain string order parameters, the short-range string correlations are calculated and so are the monotonical, oscillatory and saturation behaviors. At last, the critical exponents and central charge are extracted from the critical behaviors of order parameters around critical points and the finite entanglement of von Neumann entropy. Critical exponents ?=1 / 8 and central charge c ?0.5 indicate the universality class of QPTs of continuous QPTs line is an Ising one.In Chapter five, we make a conclusion and present our outlook for the future research.