The Study Of Dirac Equation And Topological Properties In Artificial Bandgap Materials

Recently,artificial bandgap materials(such as photonic crystals and phononic crystals)have been becoming the research hotspot of the next generation intelligent materials,because of its extremely designable,tunable and controllable capacity of classical waves.In the field of fundamental research,their band properties have drawn much attention,including flat band,Dirac linear dispersion and more band associated topological properties,which also make them a perfect platform for mimicking the physical phenomena that are hard to realize in electronic systems of condensed matter.The unique band properties lead to a number of interesting physical phenomena,such as negative refraction,beating effect,topological edge states and so on.Because of these novel phenomena,the artificial bandgap materials can be great candidates of applications in the area of optical/acoustic information and processing,even for quantum information processing,playing a crucial part in the optical/acoustic signal excitation,propagation,receiving and processing.On the other hand,Dirac equation is the basic equation to describe the relativistic particles.The basic properties of linear dispersion not only lead to fantastic physical phenomena such as pseudo-diffusion,Klein tunneling,Dirac Zitterbewegung,etc.but also represent the physical nature of topological phenomena in condensed matters.Generally speaking,the influence of the Dirac equation can be seen behind every type of topological phenomena.This thesis aims to study several kinds of artificial bandgap materials starting from Dirac equation.We deeply explore the physical essence of Dirac equation and the associated non-trivial topology phenomenon caused by lifting the Dirac degeneracy.Specifically,the Zitterbewegung effects predicted by Schrodinger according to the Dirac equation is well studied by using surface acoustic graphene structure,and the optical quantum spin Hall effect is also explored by using bi-anisotropic photonic crystals.The details are listed as follows:1.The relationship between Dirac equation and geometric phase is systematically studied,and the topological origin of condensed matter is also discussed.Then,we derive the Dirac equation from the surface acoustic wave system and the dielectric photonic crystal respectively.We further induced the Faraday gyrotropic term and effective gauge field to break the Dirac degeneracy.2.Analogue to graphene showing extraordinary electronic transport properties due to the local topological defects in bands,we proposed a particular kind of phononic "graphene",which is consist of honeycomb pillars attached on a piezoelectric material(LiNbO3)substrate.By both theoretical deduction and experimental demonstration,the Dirac like band crossing was confirmed existing at the K/K' point of the phononic"graphene" due to the symmetry protection.Around the Dirac point,there are two dramatically phenomena:(1)beating effect,which is analogous to the Zitterbewegung effect due to the interference of relative quasiparticles predicted by Schrodinger in electronic system;(2)SAW pseudo-diffusion effect,which is similar to the diffusion propagating phenomena in disorder scattering medium,but with completely difference essence.3.We try to explore the topological nature of quantum spin Hall effects.First,we propose a new photonic topological states with quantum spin Hall effect which can be realized in gyrotropic photonic crystals by using circularly polarized LCP/RCP and linearly polarized TE/TM as pseudospin to mimic the electronic spins.In such kind of topological photonic crystals,we clearly observe the one-way propagating edge state in simulation and further check the robustness of back scattering immune against to the local perturbations.Furthermore,we calculate their topological invariants-spin Chern number and Z2 invariant to clarify the band topological property of the photonic crystals.And then,under the consideration of time inversion symmetry,we clarify that a "pseudo-time inversion symmetry" is playing the key role in protecting the robustness of edge states in Bosonic topological insulators.Generally,in this thesis,I try to explore the topological nature and properties of artificial bandgap materials,starting from the Dirac equation.Specifically,in the cases of both the surface phononic crystal system and photonic crystal system,some associated exotic phenomenon,such as pseudo diffusion and extremely beating behavior around the Dirac point,optical counterpart of quantum spin Hall effects,are well theoretically and experimentally studied.I am confident that our work paves the way to accessing topological properties in artificial bandgap materials.All the results provide the basis for understanding the topological properties and transmission behavior of electrons,phonons and photons under the theory of great unity of waves.