| The ultracold atomic gas in the optical lattice is one of the ideal platform for studying quantum simulation.In recent years,there has been a great enthusiasm for the study of quantum simulation of topological states based on the ultracold atoms.The study of topological properties of ultracold atomic gases in optical lattices requires the use of specific topological invariants to characterize the topological properties of the system.In this thesis,we first review the topological studying methods,characteristics of the phase transition and related recent research progress of one-dimensional Su-Schrieffer-Heeger(SSH)model and its generalization from the perspective of tight-binding approximation.In the second part,a new realizable SSH4 model with hierarchical longrange hoppings based on one-dimensional tetratomic chains is studied.Topological states with large winding numbers can be generated in the model by selecting suitable long-range hoppings.The winding number is determined by the highest order of the long-range hopping.The positive or negative of winding number is related with different hopping directions.The boundary of phase transition between topological states with different winding numbers is controlled by intracell and intercell hoppings.In addition to the near-zero energy edge states,there exist topologically protected edge states with energy beyond zero,which are distributed at the both boundaries of the system.The number of edge states is the same as the winding number of the system under periodic boundary conditions,and the bulk-edge correspondence is exactly matched.In the third part,the SSH4 model with long-range hopping and nonreciprocal non-Hermitian term is studied,and the non-Hermitian skin effect and generalized Brillouin zone are introduced.The SSH4 model with nonHermitian high-order staggered hoppings is analyzed by using non-Bloch energy band theory.It is found that the eigenvalues of the non-Hermitian Hamiltonian are complex,and non-closed ring structures in the parameter space of the real and imaginary parts of the eigen-energy appear.Then the Hamiltonian of the system is anti-diagonalized and the generalized Brillouin zone is used to calculate the distribution of the winding number of the system under certain parameters.This work can also be extended to systems with any high-order long-range hoppings to effectively describe the topological properties of the systems.In the fourth part,the effects of the high-order staggered hoppings,nonHermitian effect and periodical driving on the topological properties of the Creutz ladder model are investigated and the possible new topological effects of the systems are revealed.Different from the SSH4 model,the trajectory diagram of the Creutz ladder with high-order and long-range hoppings is found to show multiple tangentially arranged ring structures in the energy phase diagram.The number of ring structures corresponds to the number of higherorder hoppings.Interestingly,the existence of these higher-order hoppings doesn’t affect the change of the topological invariants of the systems.In the periodically driven Creutz ladder model,the conversion of the system’s Hamiltonian into an effective Hamiltonian is realized by Floquet theory under high frequency approximation.It is found that the systems also have rich topological properties.The work in this thesis is a further extension of the study of topological properties of low-dimensional SSH-like model,revealing the novel topological properties of the system under different physical conditions,and having important guiding significance for the future experimental study of topological properties of systems such as ultracold atoms in optical lattices and classical circuits. |
