Topological States In Ultracold Atomic Gases: Unconventional Lattices And Interaction Effects

In this thesis,we describe three research projects on topological states in ultracold atomic gases.In the first two studies,we propose respectively two experimental schemes to synthesize two-dimensional topological lattices for ultracold atoms.In the third one,we investigate in detail the effect of interactions on the one dimensional topological superconductor(or a Kitaev chain)realized using polarized dipolar fermi atoms.In the first project,we propose a scheme to dynamically synthesize a spaceperiodic effective magnetic field for neutral atoms by time-periodic magnetic field pulses.When atomic spin adiabatically follows the direction of the effective magnetic field,an adiabatic scalar potential together with a geometric vector potential emerges for the atomic center-of-mass motion,due to the Berry phase effect.While atoms hop between honeycomb lattice sites formed by the minima of the adiabatic potential,complex Peierls phase factors in the hopping coefficients are induced by the vector potential,which facilitate a topological Chern insulator.With further tuning of external parameters,both topological phase transition and topological quasiflat band can be achieved,highlighting realistic prospects for studying strongly correlated phenomena in this system.In the second project,we propose a scheme to dynamically generate an optical flux lattice with nontrivial band topology using amplitude modulated Raman lasers and radio-frequency coupling fields.By tuning the strength of Raman and radio-frequency fields,three distinct phases are realized at unit filling for the unit cell.These three phases respectively correspond to: normal insulator,topological Chern insulator,and semi-metal.Nearly nondispersive bands are found to appear naturally in the topological phase.The validity of this proposal is confirmed by comparing the Floquet quasi-energies from the evolution operator with the spectrum of the effective Hamiltonian.In the third project,we describe a scheme to realize a one-dimensional Kitaev's p-wave superconducting chain model with interaction using polarized dipolar atomic fermi gases.We use exact diagonalization method to numerically calculate the energy spectrum of this interacting system,and establish the many-body ground-state phase diagram.The phase diagram includes five phases: the insulator phase at unit filling,the vacuum phase at zero filling,the Mott insulator phase at half filling,the topological superconductor phase,and the Luttinger liquid phase.We use ground-state energy gap,fidelity susceptibility,number parity,and compressibility to describe the properties of the ground states in different parameter regimes.The Majorana zero mode wavefunctions with interaction are calculated numerically.We analyze the entanglement properties of the ground states by calculating the entanglement spectrum and entanglement entropy as functions of interaction strength.The central charges at the critical points are also extracted.The gapless nature of the Luttinger liquid state is also verified through the finite size scaling method.