Phase Transitions In One And Two Dimensional Spin Lattice Quantum Many-body Systems: Fidelity Approach

This thesis studies quantum phase transition in strongly-correlated many-bodysystems in the perspective of quantum information, such as fidelity and entanglement.Rencent development of numerical approaches based on tensor networks allow us toimprove our understanding on fundamental aspect of quantum phase transitions, as wellas on exploring newly found exotic quantum phase, for instance, topologicaly orderedstate.The first chapter of this thesis is the introduction, mainly introducing thebackgroud related to this subject. It begins with the definition of some basic concepts,such as quantum phase transition, thermal phase transition, spontaneous symmetrybreaking, order parameter and fidelity. Then it presents area theorem and tensor networkstates. At last it introduces some tensor network algorithms based on the tensor networkrepresentation, in which both the wave function and environment updating process areexplained. The main part of the thesis is presented from chapter two to chapter five.In Chapter two, it mainly in detail discuss the ground state fidelity phase diagramof the Heisenberg antiferromagnetic chain with exchange and single-ion anisotropies.First, the phase boundaries between each of two phases are determined by the groundstate fidelity per lattice site. The reason why fidelity can determine phase boundaries isthe pinch point occurrs on the ground state fidelity surface. Corresponly to the quantumphase transition points. Meanwhile, it is found that a magnetization plateau correspondsto a fidelity plateau on the ground state fidelity surface, this offers an alternative routetowards the investigation of the magnetization processes of quantum many-body spinsystems. In addition, the central charges at criticalities are identified by performing afinite entanglement scaling analysis. The results show that all the phase transitionsbetween the spin liquids and magnetization plateaus belong to the Pokrovsky-Talapovuniversality class. Finally, this chapter introduces how to extract parameter K by MPSalgorithm with U(1) symmetry.In Chapter three, it mainly introduces the investigation of the two-dimensionalq-state quantum Potts model on the infinite square lattice by using the infinite ProjectedEntangled-Pair State(iPEPS) algorithm. At the very beginning, it tells how to detectq-fold degenerate ground states for the Zq broken-symmetry phase by using quantumfidelity which is defined as an overlap measurement between an arbitrary reference state and the iPEPS ground state of the system. Similar to the characteristic behavior of thequantum fidelity, the magnetizations, as order parameters, obtained from the degenerateground states exhibit multiple bifurcations at critical points. In addition, this chapterintroduces the universal order parameter of q-state quantum Potts model. At last itshows that the critical point of the two-dimensional q-state quantum Potts model undercontinuous limit can be obtained by extrapolating the critical points corresponding todifferent q with q gong to the infinite.In Chapter four, it introduces the quantum phase transitions of different models onthe honeycomb lattice. Ising model, XXZ model and Kitaev-Heisenberg model aresimulated respectively by adopting simple update iPEPS algorithm. The order parameterand ground state fidelity of these three kinds of model are also investigated respectively.Moreover, for Ising model its bifurcation of the ground state fidelity is studied, inwhich a fidelity bifurcation point is detected as the phase transition point.In Chapter five, it mainly introduces the thermal phase transition ontwo-dimensional Archimedes lattice. Firstly, it introduces the tensor network algorithmof projected entangled pair states with ancillas at finite temperature implemented withimaginary time evolution on the honeycomb lattice. Secondly, the Ising and XXZmodels with both zero field and finite transverse field are simulated. From the surface ofthe magnetization, different phase transiton temperatures can be read directly. Moreimportantly we investigate fidelity per lattice site finite temperature for the quantumIsing model on the honeycomb lattice, from which we prove that the fidelity per latticesite can detect the thermal phase transiton point, successfully. At the end of this chapter,some simulation results about Ising model at finite temperature on square lattice arepresented.In Chapter six, we draw the conclusion and present our proposed for futureresearch.