Critical Phenomena In Two-and Quasi-one-Dimensional Quantum Systems

There is a very important physical problem in condensed matter physics, which is,at the absolute zero temperature and for some non-thermodynamic control parameters(for example, magnetic field, chemical composition or pressure, etc.), that how to findthe ground-state energy of a quantum system. It indicates that quantum phase transitions(QPTs) may also occur in some materials. QPTs caused by quantum fluctuations are dueto Heisenberg’s uncertainty principle. For the QPTs, a very interesting feature is theexisting of the critical phenomena in quantum systems. Such quantum criticalphenomena in condensed matter systems, in particular, the physical properties of theinteracting many-body system play an important even crucial role.In Sec. Ⅰ, it introduces the QPTs and quantum critical phenomena, and explains thetensor network (TN) algorithms and the ground-state fidelity theory. The ground statefidelity can be used to quantify the QPTs in quantum systems, regardless of the internalorder is quantum symmetry breaking order or novel topological order. This alsoillustrate that the fidelity theory provides an important perspective in quantuminformation and condensed matter physics.In Sec. Ⅱ, the infinite projected entangled-pair state (iPEPS) algorithm fortwo-dimensional quantum spin systems is briefly discussed. We establish an intriguingconnection between the QPTs and the ground-state fidelity per lattice site, and constructthe universal order parameter for quantum Ising model on an infinite-size square latticein two spatial dimensions with symmetry breaking order. This is achieved by computingground-state wave functions in the context of the TN algorithm based on the iPEPSrepresentation. The ground-state fidelity per lattice site is computed for quantum XYXmodel on an infinite-size square lattice in two spatial dimensions. It demonstrates thatthe field-induced quantum phase transition is unambiguously characterized by a pinchpoint on the fidelity surface, which marks a continuous phase transition. We alsocompute an entanglement estimator, defined as a ratio between the one-tangle and thesum of squared concurrences, to identify both the factorizing field and the critical point,resulting in quantitative agreement with quantum Monte Carlo simulation. In addition, abifurcation in the reduced fidelity between two different reduced density matrices andthe local order parameter are also discussed.In Sec. Ⅲ, a TN algorithm is developed for spin ladders, which can generate the ground-state wave functions efficiently for infinite-size quantum spin ladders. Thealgorithm is also able to efficiently compute the ground-state fidelity per lattice site,which is a universal phase transition marker. Thus it offers a powerful tool to unveilquantum many-body physics underlying spin ladders. To illustrate the scheme, weconsider the two-leg and three-leg Heisenberg spin ladders with staggering dimerization,the two-leg Heisenberg spin ladder with cyclic four-spin exchange, and theferromagnetic frustrated two-leg ladder. The ground-state phase diagrams thus yieldedare reliable, compared with the previous studies based on the the exact diagonalization(ED) and the density matrix renormalization group (DMRG). The results indicate thatthe ground-state fidelity per lattice site successfully captures quantum criticalities inspin ladders.In a word, this thesis has developed efficient TN algorithms for the simulations oftwo-dimensional and quasi-one-dimensional quantum systems. Through such efficientTN algorithms, the ground-state fidelity per lattice site is able to study the QPTs andquantum critical phenomena in two-dimensional and quasi-one-dimensional quantumsystems. This is manifested that the ground-state fidelity per lattice site can exhibit apinch point in the three-dimensional surface to detect the QPTs. Furthermore,researches for the QPT in two-dimensional and quasi-one-dimensional quantum systemsalso have shown that, quantum information theory provides a general method for theQPT: when the local order parameter or non-local order parameter (order parameter isassociated with the system model of a physical quantity) in two-dimensional andquasi-one-dimensional quantum many-body systems is unknown, or whether there isorder parameter, the quantum critical point can be found through the ground statefidelity per lattice site.