Topological Phase And Topological Phase Transition With Ultracold Atoms

Ultracold atoms are presented as an excellent platform for simulating topological phases and topological phase transitions due to its high feasibility with manipulation by quantum optical methods.Recently,the developing methods on spin-orbit coupling,shaking optical lattice,laser assisted tunneling have paved way for simulating topological phases.Considering this,the dissertation is aimed to study topological supersolid states in non-equilibrium dipolar system,Robust Majorana edge modes in driving system,as well as topological phase transition,topological exciatations in other equilibrium system:?1?Achieving topological supersolid states with dipolar Fermi gases trapped in a spin dependent optical lattice:Recently,there has been previous study on realizing supersolid states with dipolar Fermi gases due its anisotropic features.However,such supersolid states is trivial in topology.Therefore,we consider to combine p_x+ip_y topological superfluid order with stripe order to synthesize the topological supersolid states.When the dipole moments tilt towards a proper orientation,its interaction can be projected to balanced attractive parts in separate directions,leading to p_x+ip_y topological superfluid states.However,traditional stripe order originates from repulsive dipolar interaction,and does not apply to our system.To regain the stripe order,we tend to utilize the spin dependent optical lattice.Numerical mean field calculation exhibits that stripe order can be obtained by the cooperation of dipolar coupling and staggered hopping scenario.Meanwhile,above stripe order caninteraction strength,which leads to topological supersolid states.As the supersolid states are gained with attractive interaction,we need to prove the system is stable against collapsing.Considering this,the inverse compressibility is calculated,of which the positive sign denotes the stability.To the end,we propose how to simulate the staggered next-next nearest hopping via laser assisted tunneling,which is equivalent to spin dependent optical lattice.?2?Achieving Robust Floquet Majorana edge modes in periodically drivensystem:Periodically driven system features fruitful topological properties due to the non-equilibrium.There has already been previous study on driven Kitaev chain.However,it can be seen that Floquet Majorana edge modes can be hard to be recognized,and can easily merge to the bulk states with low driving frequency.To overwhelm the problem,we consider a dynamic Kitaev chain with multiple time periodic driving term.Numerical results show that multiple driving can generate more gaps in Floquet spectrum,leading to more robust Floquet Majorana edge modes.Strictly,with the enhance of multiple driving,the tunneling between edge modes will be suppressed,and edge modes become more localized.To identify the Floquet Majorana edge modes from bulk invariant,we tempt to define a Zak phase based on the Floquet bands in frequency space.The vanishing of the Zak phase decides whether it is topologically nontirvial.Finally,we find that no degenerate points of phase bands at high frequency limit,and analytical form of time independent effective Hamiltonian can be obtained.Our results show that the relative phase in multiple driving play a dominant role in influencing topological phase transitions.Above multiple driving process can be simulated via Raman transitions.?3?We also study the topological phase transtions and topological exciatations in Other equilibrium system.We demonstrate that the multicomponent can be viewed as impurity.When the reative chemical potential increases,the Majorana can be combined to the fermion mode,which is induced by the impurity.We study the honeycomb lattice with high orbital tunneling,and find threefold degenerate fermions.Honeycomb lattice possess C_3vsymmetry,and can have more bands consierding higher orbitals.The threefold degenerate points are stable against changing the amplitude of tunneling.