Topological Properties Of Low-dimensional Quantum Systems

The research on the topological properties of the system is one of the hot topics in condensed matter physics.The tight binding approximation is an effective method to help us understand the topological properties of the sys?tem at the theoretical level.In the theoretical framework of the tight binding approximation,we first discuss the topological properties of one-dimensional Su-Schrieffer-Heeger(SSH)model from the perspectives of winding number,geometric representation and bulk-edge correspondence.We know that the one-dimensional simple tight binding model describes the trivial metal with a single band,where there is no any topological phenommenon.If there is particle pairing,then such a system may be a topological superconductor.Topological superconductor is a kind of superconducting system protected by topology,and Kitaev model is the first tight binding model to interpret the topologi-cal properties of the topological superconductor.We discuss the relationship between the topological superconducting phase transition and the Anderson localization phase transition of a one-dimensional extended Kitaev model by physical quantities such as band gap(Gap),mean inverse participation ratio(MIPR),and topological number(Z2).The former topological models we men-tioned are static models.However,topological phenomena do not only occur in static systems.If the system is periodically driven,it will also have non-trivial topological properties.Moreover,periodic driving is a flexible and an efficient method for generating Floquet topological states,and this method has been widely researched.By this method,Haldane model and Hofstadter model were successfully implemented in cold atom experiments.Inspired by this,within the framework of tight binding approximation,we have designed a period-driven quantum protocol based on the Dice lattice(extension of one-dimensional compound lattice in two-dimensional space).In this system,we find the topological nontrivial phases with large Chern number(C).This mas-ter thesis contains the following five chapters:In chapter 1,we introduce the research background of the condensed mat-ter topological systems,and reviews some involved physical knowledge,such as energy band,tight binding model and discrete Fourier transform.In chapter 2,we analyze the topological properties of the one-dimensional SSH model.Firstly,we deduce how to construct the Hamiltonian for this type of compound lattice.Through the periodic boundary energy spectrum,we know that the system has the characteristics of an insulator.After the Fouri-er transformation,we have the Hamiltonian representation of the momentum space.Then,we analyze the trajectories of Hamiltonian parameters and topo-logical invariants-winding number,and we can determine that the system has three quantum phases:topologically nontrivial bulk insulating phase,topolog-ically trivial insulating phase,and the metallic phase.Finally,we analyze the probability distribution of wave functions in different phase regions and verify the bulk-edge correspondence.In chapter 3,we analyze the topological properties of the one-dimensional extended Kitaev model.The relationship between topological superconduct-ing phase transition and Anderson localization phase transition is discussed by means of Gap,MIPR,and Z2.In chapter 4,we study the topological properties of two-dimensional dice optical lattices model under periodic driving.After calculation,we find the topological nontrivial phases with large C.The phase diagrams are verified by the edge-state quasienergy spectra.In chapter 5,we make conclusion and outlook.