Hierarchical Topological Phenomena Protected By Nonsymmorphic Symmetries In 2D Phononic Crystals

The rapid development of topological physics began with the discovery of the quantum spin Hall effect.The quantum spin Hall states possess time reversal symmetry-protected gapless boundary states,which have the characteristics of spin-momentum locking.For acoustic or optical systems without spin degrees of freedom,we can construct specific pseudospin and pseudo time reversal symmetry that satisfies T2=-1 with additional degrees of freedom,similarily,pseudospin-momentum-locked boundary states can be obtained.In recent years,the search for higher-order topological materials has become a research hotspot in condensed matter physics.Higher-order topological insulators reveal a new bulk-edge correspondence,that is,the dimensions of the boundary states are lower than those of the traditional topological insulators of the same dimension.The quadrupole topological insulator belongs to a two-dimensional second-order topological insulator with topologically protected zero-dimensional corner states.Quadrupole topological insulators rely on π-flux and anticommutative mirror symmetries,which are difficult to achieve in solid materials.Therefore,the current experimental realizations of quadrupole insulators are based on artificial periodic systems.In this paper,we use the artificial periodic phononic crystal in two-dimensional square lattice as the research system.Within such the system,we realized the quantum spin Hall states and anomalous quadrupole topological states,respectively,in the first and second band gaps.For the first band gap,the nonsymmorphic glide symmetries are used to construct the pseudo time reversal symmetry,and we adjust the rotation angle of the structural primitives so that a topological phase transition occurs with odd and even parities of eigenmodes switching at M point,then,the quantum spin Hall states of sound are obtained.For the second band gap,we also utilize the nonsymmorphic glide symmetries to replace the anticommutative mirror symmetries required by the canonical quadrupole insulators,then,the anomalous topological quadrupole corner states are achieved.By adjusting the geometric parameters,the anomalous quadrupole states can also undergo a topological phase transition,which is characterized by the closure of the Wannier bands.The topological states of the two band gaps are both observed experimentally,and the robustness is verified.We have achieved the novel hierarchical phenomena of coexisting first-and second-order topological states in the same acoustic system,which is missing in solid materials with Fermi energy levels.The acoustic topological states are expected to be widely used in various aspects such as acoustic resonators,directional radiation of sound,acoustic isolation,integrated sound devices,and so on.