| Topological quantum systems,including topological superconductors,topological insulators,topological nodal systems,etc.,have many novel properties which have attracted many attentions,and it can be used in the quantum information theory,developing the special quantum devices in the future,etc.Topological quantum systems have the edge states which are protected by topological properties,such as the quasi Majorana Fermion at the end of one dimensional topological superconductor.the insulating bulk states and conductive surface states in two dimensional topological insulators,etc.Topological quantum systems can be characterized by corresponding topological invariants,which indicate different topological states.Among the researches of topological systems,the one-dimensional Su-SchriefferHeeger(SSH)chain has a simply form and structure,with strong representation of topological properties.Furthermore,SSH model can be used to simulate high dimensional topological systems,or study the other topological systems which can be mapped to it.In experiments,SSH model can be realized in the polyacetylene lattice,and the interaction parameters can be controlled easily.Thus,the deep study of the SSH model is more useful in further experiment research.In recent years,it has been shown that the periodic driving can modulate the topological properties of a system,and new topological phases can be obtained by setting driving parameters,which can be easily realized in experiments.Based on these researches,we mainly studied the properties of some extended SSH models.This thesis has been divided into seven chapters,and our main research work have been included from Chapter 3 to Chapter 6.In Chapter 1 and Chapter 2,we give some introductions of the basic knowledge,such as the background of quantum mechanics,topological quantum systems,periodic driving,etc.In Chapter 3,the cyclically modulated SSH chain systems with long range hoppings,including the nearest-neighbor,the next-nearest-neighbor and the third-nearest neighbor hoppings,have been proposed and investigated.The topological nontrivial phases characterized by Chern number have been obtained,showing a rich phase diagram in the two-dimensional space of the momentum and the modulation parameter.A similar shape of phase diagram but with different Chern numbers with Haldane model has been shown,and higher Chern numbers have been obtained and verified,indicating more edge states can be obtained in such system.In Chapter 4,topological properties of an extended SSH model under the periodic ?function kick with X-direction,Y-direction,and Z-direction defined by pseudospin expression of the Hamiltonian in momentum space,has been explored.We find that,by modulating driven parameters and periodic ?-function kick in such extended system,fruitful phase diagrams and topological states with higher Chern numbers can be introduced.In the case of X-direction kick and Z-direction kick,topological phase diagram will be changed but Chern numbers remain 0 and ±1,while for Y-direction kick,large Chern numbers ±2 can emerge.This is an extended study of using periodic kick to obtain fruitful topological phases and large Chern number states in simulated two-dimensional systems.In Chapter 5,topological properties of the SSH model with a long distance interaction under the periodic ?-function kick has been explored and topological phases with larger Chern number ±4 can be obtained in the X-direction kick.In Chapter 6,we studied several non-Hermitian SSH systems with different coupling methods,including a non-Hermitian coupled SSH chain system,two coupled non-Hermitian SSH chain systems,and two non-Hermitian coupled non-Hermitian SSH systems.In the case of periodic boundary condition,we have given the phase diagrams of the winding number of these three models respectively,and found that different coupling methods will affect the value of the winding number and the shape of the phase diagram.In Chapter 7,we summarize our research work. |
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