Acoustic Floquet Topological Insulators

The magic one-way edge states,existed at the interfaces between ordinary insulators and electronic topological insulators,are immune to various defects,which may have potential applications in the lossless energy transfer and information communication.Thanks to the similarity between the Schr?dinger equation for electrons and the Helmholtz equation for classical waves,the paradigm shift has been established in recent two decades for topological transports,from electronics to photonics,mechanics and elastics,electric circuits,airbone and underwater acoustics,which has become a hotspot in the field of condense matter physics.The mechanisms for vairous classical-wave analogies include(a)the use of gyromagnetic materials or constructing the effective magnetic field to generate chiral edge states,(b)the use of band inversion and pseudo-spin coupling to generate the spin/valley-depedent edge states.Besides,the time or space modulated Floquet lattices,which has abundant physics,could be adopted to mimic the quantum Hall effect or the quantum spin Hall effect.In this thesis,we will comprehensively introduce the spatially modulated acoustic Floquet systems in different dimensions,and demonstrate the inherent topological phase transitions and topologically protected sound transport.Firstly,we explore the necessary conditions for the edge states in one-dimensional cavity chains and the physical mechanisms for the topologically protected bound states in one-dimensional spatially modulated waveguide systems.Though the diatom chains and dimer chains do share similar bandgaps,only the dimer chains will support edge states on solitary ends,where the coupling strength in the terminal unit-cells is weak.The topological invariant for those systems,namely winding number,is nonzero only for the intra-coupling strength being larger than the inter-coupling strength.Furthermore,we explore the sufficient condition for the existence of topologically protected bound states in one-dimensional dimer waveguide arrays which are spatially modulated along the propagation direction.Secondly,we for the first time demonstrate the acoustic Floquet topological insulators and pseudo-spin-dependent edge states in experiments.We calculate the projected band structures at different coupling strengths based on the scattering matrix analysis in the two-dimensional metamaterial ring arrays,unveiling that the pseudo-spin-dependent edge states survive only for the coupling strength above the threshold of?/4.The paired edge states carrying reversed pseudo-spins have opposite group velocities along the boundary of the truncated lattice.Topological edge states are robust against the backscatterings after passing through the exsiting defects on the path.When we put together two Floquet lattices with half period dislocation,we will observe topological interface states.Besides,we have measured the field distributions of edge states,bulk states and band-gap excitation,which agrees with the calculated dispersive projected band structure.The criterions for distinguishing the edge states,bulk states and band gaps are provided.For boson-like systems T~2=1,the pseudo-spins for acoustic waves are not completely decoupled from each other.In experiments,we utilized strong couplings between neighboring rings to prohibit the pseudo-spin flipping.Thirdly,we propose the static three-dimensional Floquet lattices,which supports chiral edge states on the surfaces with topological protection and cyclotron orbiting states in the bulk,showing a stark resemblance to the quantum Hall effect.In the previous studies,effective magnetic fields in acoustics are created by using angular momentum bias from circulating airflow.However,the precise control of the synchronous rotation and the stability of the airflow is quite challenging.Here we propose the three-dimensional acoustic Floquet lattice with screwed coupling arms,where a time-space mapping is judiciously employed.The static configuration greatly reduces the fabrication difficulty.Robust surface states are demonstrated in the three-dimensional Floquet lattice with simulations,which agree well with the prediction of the Floquet theory.Reversing the chirality of the screwed coupling arms directly flips the group velocity direction of the edge states.Splicing two Floquet lattices with opposite chirality,we demonstrate broadband and lowloss topological negative refraction on the crystal surface.