The Topological Excitations Of Bosonic Superfluid In The Optical Lattice

This master's thesis is a summary of the author's study and work during the master's degree.In this paper,with the help of the ultracold atomic optical lattice as the excellent platform for quantum simulation,we propose several experimental schemes to realize the topological nontrivial elementary excitations for bosonic atoms,and attempt to summarize the universal law about the generation of bosonic nontrivial topology.This provides theoretical guidance for the future search for other bosonic topologically nontrivial systems,and then explores the physical essence behind the bosonic nontrivial topology.This article first briefly introduces the related theory of the Wannier function,including the definition and construction of the Wannier function.This function is the basis for obtaining the theoretical model of the bosonic system in the optical lattice;next,we introduce the Bogoliubov theory which characterizes the elementary excitation of bosonic atoms,including the construction of the bosonic Bd G Hamiltonian and the method of analyzing its topological properties;then,we study the properties of the quadratic band crossing point in the energy spectrum,we find that this particular degenerate point is closely related to the topology of the bosonic atoms;finally,we propose three examples that realize bosonic nontrivial topology,including a square lattice,a hexagonal lattice,and a boron nitride lattice.Based on the analysis of related theories and the verification of concrete examples,this paper proposes a general strategy and law for generating topologically nontrivial excitations in ultracold bosonic atom systems: when the ground states of the single-particle part of the Hamiltonian are in the degenerate space which is formed by a set of time-reversal bases,after the special ferromagnetic repulsive interaction being added,the degeneracy of the ground state of the total Hamiltonian will be eliminated,in this process,the time-reversal symmetry will be broken.If there are degenerate points protected by time-reversal symmetry in the energy spectrum,e.g.,Dirac points or quadratic band crossing points,these degenerate points will be opened and generate the energy gaps with the nonzero Berry flux,and then the topologically nontrivial energy bands will appear in the excitation spectrum.