Study Of Topological Properties In Periodically Driven Multiparticle Quantum Systems

In the past decades,topological phases in condensed matter systems have been studied extensively,most of the studies on topological phases such as topological insulators,topolog-ical superconductor were focused on static systems.However,following the study of topolog-ical materials,stimulating us to find and engineer the system with new topological phases.People find that periodic external driving topological quantum systems can have more novel properties than the static ones,even can turn nontopological materials into topological ones,thus the periodic driving has become an important method to tune the topological quantum systems.The Floquet theory can be used into the periodic driving quantum systems,There have been various novel phases in Floquet systems including Floquet topological insulators,Floquet topological superconductors and Floquet Weyl semimetals.Based on the above re-search background,in this thesis,we mainly study the topological properties of the Chern insulator and a particular p-wave topological superconductor under the periodic driving.1.We have studied the ?-function periodic kicking on the QWZ Chern insulators,and discussed the influence on the topological properties,this method establishes a class of exactly solvable systems,we have shown that,by adding the periodic kicking on a normal topological quantum system,one can also obtain the effective long-range hopping between the sites,which can generate multiple Dirac cones in the effective Hamiltonian,thus,a rich phase diagram and large Chern numbers can be obtained and tunable by this special kind of Floquet periodic driving.These results are useful for further understands or applications of Floquet topological phases,and also provide a wide way to control topological quantum systems.2.We have studied the Kitaev model with nearest-neighbor and next-nearest-neighbor interactions and a periodic driving in chemical potential,we consider two kinds of the driv-ing case,i.e.,the periodic cosine driving and the ?-function kicking.We obtain the effective Hamiltonian,which can be similarly analyzed as the Hamiltonian in static systems.A rich phase diagrams and topological phase transitions have been obtained by controlling the ac field,that should give rise to signature of Majorana states in experiments,the time depen-dence of these Majorana states allows us to manipulate them,our work contains the driving in different frequency regimes.In our periodic ?-function kicking,This method gives an ex-actly solvable system,we can get rich topological phases and have many Majorana modes by modulating the kicking parameters and the next-nearest-neighbor hopping amplitude.Many Majorana modes presented at the edge provides a favorable platform for the signature of Majorana fermions in experiments.3.We have numerically studied the Kitaev model with nearest-neighbor and next-nearest-neighbor interactions.We show that through the periodic driving of the pairing potentials,two types of end modes can also be generated in our model,i.e.,the conventional end modes and anomalous end modes.We have numerically studied the driving of the first-neighbor and second-neighbor pairing potential,and also analyzed the end modes,Floquet eigenvalues and the Fourier transform of these end modes,these results are useful for further understanding of the bulk-boundary relationship and searching edge modes in experiments.