Localized States And Topological States Of One-Dimensional Quasiperiodic Systems

In this dissertation,I have studied the localized states and topological s-tates of one-dimensional quasiperiodic systems.We firstly discuss the localized properties of Aubry-Andr?e?AA?model when the wave vector is small,when adding p-wave pairing or adding the nearest-neighbor interaction.We then study the phase transitions of a Weyl semimetal?WSM?model when adding an incom-mensurate potential in one direction of the three-dimensional system?it can be reduced to a one-dimensional?1D?system?.Finally,we investigate the charac-terization of topological properties of dimerized Kitaev chains by introducing two edge correlation functions.In chapter 1,I will first make a simple introduction to the 1D AA model and many body localization?MBL?.After that I will discuss the symmetries of the topological states and some topological invariants.Finally,I will make a pedagog-ical introduction to three 1D topological models:the Su-Schrieffer-Heeger?SSH?model,Kitaev model and 1D p-wave superconductor model with the incommen-surate potential.In chapter 2,We discuss the AA model when the wave vector is small and when adding p-wave pairing.It is well known that the eigenstates of the AA model are either extended or localized depending on the strength of incommensurate potential V being less or bigger than a critical value V_c,which equals to double the hopping strength and no mobility edge exists.However,when the wave vector?is very small???1?,this model shows different properties.By using the inverse participation ratio?IPR?and performing the multifractal analysis,we find that the existence of mobility edges E_c±and E_c.±for the AA model with V<V_c,and the eigenstates are divided into three different regions by these mobility edges.The eigenstates of the system are critical when|E_c±|?|E|?|E_c?_±|,they are extended when E_c-<E<E_c+and they are localized when E<E_c?_-and E>E_c?₊.The AA model with p-wave superconductor pairing can be used to study the transition from topological superconductor?TSC?phase to Anderson localization induced by the incommensurate potential.Actually,there exist three regions in the phase diagram of this system:the extended,critical and localized regions.we study the spectral statistics of the three regions and find that the bandwidth distribution and the level-spacing distribution display an inverse power law?IPL?in the critical region.We then perform a finite-size scaling analysis on system's wavefuntions to determine critical exponents of the system in the critical region and unveil the existence of a hyperscaling law among these critical exponents.Finally,we carry out a multifractal analysis on the wavefuntions of this model.In chapter 3,We study the localized properties of AA model with the nearest-neighbor interaction.By calculating the normalized participation ratio and the distribution of nature orbitals of the one-particle density matrix,we unveil that the phase diagram of this system is composed of three different phases:an er-godic phase,a MBL phase and an intermediate phase.We determine the phase boundary between the ergodic and intermediate phases by studying the density distribution of the single particle excitation by using the density matrix renor-malization group and estimate the boundary between the intermediate and MBL phases by studying energy level statistics and the entanglement entropy.In chapter 4,We study the effect of an incommensurate potential is added on one direction of the WSM.The momentums of the other two directions are good quantum numbers,so this system can be reduced to a 1D model.By studying the IPR of wave functions and using the properties of AA model,we investigate the extended-localization transition of the system in the direction of the added potential.We then demonstrate the occurrence of a transition from WSM to a two-dimensional metallic phase by analyzing the change of density of states of this system.In summary,there exist WSM-localized in the direction of the added potential-2D metal quantum phase transitions when the strength of the incommensurate potential increases.In chapter 5,we introduce two edge correlation functions to characterize d-ifferent topological phases of an noninteracting and interacting dimerized Kitaev chain,which includes three different phases,i.e.,the trivial,SSH-like topological,and TSC phases,in different parameter regions.We identified the phase dia-gram by using two different methods by calculating the topological number under periodic boundary conditions and two introduced edge correlation functions of Majorana fermions under open boundary conditions,respectively.Comparing these two methods,we find the latter method is more intuitive and gives directly the coupling information for the edge Majorana fermions.When adding the in-teraction,we can also analytically calculate the two introduced edge correlation functions at some symmetric points,which can also be used to characterize the different topological phases of the interacting system.