| The discovery of quantum Hall effect opened the door to the study of topological phase of matter.In the last 40 years,topological phases have been explored not only in electronic systems,but also in various artificial periodic structures.The artificial periodic structure system can simulate the energy band,eigenstate and edge state of the electronic system with the same Parity-Time symmetry,which is an effective tool to study the topological properties of matter.So far,topological phases can be simply divided into first-order topological phases and higher-order topological phases,which have been proved theoretically and experimentally.In this paper,topological insulators in two-dimensional Kagome system are studied,including hybrid-order topological insulators in a phononic crystal and the topological phase transitions between first-order and higher-order topological insulators in phononic crystals and circuits.The main works are listed below:1.We proposed a new kind of hybrid-order topological insulator in a two-dimensional bilayer Kagome phononic crystal,which can realize the coexistence of different topological phases in the same system.To be specific,different modes of air cavities in acoustic systems play different roles.The monopole mode coupling induced the first-order topological phase,and the dipole mode coupling induced the higher-order topological phase.Different topological phases are realized in the band gaps with different modes.The first-order topology is characterized by a pair of one-dimensional gapless helical edge states with counterpragrate direction.Topological invariant is spin Chern number.The second-order topology is described by a pair of zero-dimensional corner states with in-phase and anti-phase pressure distributions at a corner.Topological invariant is spin polarization two-dimensional Wannier center.By measuring the sound pressure fields of the experimental sample,the helical edge states and zero-dimensional corner states are observed experimentally.The existence of hybrid-order topological insulator is confirmed by numerical simulation and experimental measurements.2.We systematically proposed the topological phase transitions between spin-Chern insulators and second-order topological insulators in the bilayer Kagome tight-binding model and phononic crystals.The structure is constructed by a layered stack design,in which two single-layer Kagome models are stacking with different nearest neighbor coupling within the intralayers and chiral helical coupling between the two layers.The competition of these two coupling effects results in a topological phase transition from a spin-Chern insulators to a second-order topological insulator,characterized by a pair of one-dimensional helical edge states on the boundary to a pair of zero-dimensional corner states on the corner.Furthermore,the tight-binding lattice model is mapped to phononic crystals,and the same topological phase transition is achieved in phononic crystals with the diameter parameters change of the coupling tubes.We use 3D printing technology to make experimental samples and verify the existence of topological helical edge states and topological corner states.3.We systematically studied the topological phase transition between ordinary insulator,quantum spin Hall insulator and second-order topological insulators in a two-dimensional Kagome circuit.The LC circuit is used to construct the system,and capacitance(（6） or(（7） is introduced between the nearest neighbor sites to construct the unequivalent nearest neighbor coupling effect,which can induce the second-order topological state.The equivalent spin-orbit coupling,which induce the quantum spin Hall states,is constructed by introducing capacitance(（8） between the next nearest neighbor sites.By analyzing the circuit through Kirchhoff’s current law and Ohm’s law,the Laplacian matrix and the Hamiltonian of the circuit can be obtained.The topological phase transition between the second-order topology and the first-order topology is theoretically analyzed under the competition of two coupling interactions.We also observe the zero-dimensional corner state in the second-order topological system and the one-dimensional helical edge state in the first-order topological system experimentally,and verify the existence of various topological phases in the phase diagram. |
