Topological Edge States of Concave Hexagonal Gyroscopic Phononic Crystals

doi:10.16553/j.cnki.issn1000-985x.2025.0120

Abstract

Abstract: The research on gyroscopic metamaterials has pioneered a novel approach for topological acoustics. By incorporating gyroscopic structures into an infinitely periodic discrete medium with a concave hexagonal lattice， a gyroscopic phononic crystal （GPC） is proposed， with analysis focusing on the propagation of torsional waves with this structure. This work examines the bandgap characteristics of the gyroscopic phononic crystal， while exploring the mechanisms behind the opening of Dirac cones and the emergence of topological edge states induced by variations in gyroscopic torque. Subsequently， the influence of the gyroscopic rotation speed on the bandgap was studied in detail， and phenomena such as band inversion and Hall effect were discovered. It is shown that by breaking both structural symmetry and time-reversal symmetry， topological edge states can be identified near both of the opened bandgaps in the same topological gyroscopic metamaterial. The research is extended to analyze the unit cell of the topological gyroscopic metamaterial， discussing the wave propagation properties of topological interfaces within the two newly formed bandgaps under different configurations. This reveals directional disparities in the topological edge states between the upper and lower bandgaps of the topological gyroscopic metamaterial. Additionally， the robustness of these topological edge states against defects in the gyroscopic metamaterial is demonstrated.

Key words: gyroscopic phononic crystal; metamaterial; Dirac cone; topological edge state; wave propagation

CLC Number:

XIAO Weimin, NIE Jingkai, ZHAO Junjuan, HU Wencheng, HAN Yu, SHI Lei. Topological Edge States of Concave Hexagonal Gyroscopic Phononic Crystals[J]. Journal of Synthetic Crystals, 2025, 54(11): 1937-1946.

Figures/Tables 13

Fig.1 （a） Schematic of unit cell of the concave hexagonal GPC structure connecting with springs；（b） schematic of the concave hexagonal lattice

Fig.2 Band structure of the GPC with λ=0 and λ=1/3

Fig.3 Relationship between the bandwidth and λ

Fig.4 Topological edge state band structure （a） and schematic of the supercell and displacement fields （b）

Fig.5 Gyroscopic metamaterial with a topological interface and displacement fields with 451 Hz at different times

Fig.6 Wave transmission rate in the column of excitation application points at the lower band gap

Fig. 7 Gyroscopic metamaterial with a topological interface and displacement fields with different frequencies

Fig.8 Amplitude in the column of excitation application points near the upper boundary of the upper band gap

Fig.9 Gyroscopic metamaterial with a topological interface and displacement fields with frequencies at the lower band gap. （a） GPC with type B upside and type A downside，（b） 81 Hz，（c） 95 Hz，（d） GPC with type A upside and type B downside with 95 Hz

Fig.10 Gyroscopic metamaterial with nine cells and displacement fields with frequencies on the edge of the upper band gap. （a） GPC with nine cells combined with types A upside and B，（b） 450 Hz at node Ⅰ，（c） 450 Hz at node Ⅳ，（d） 352 Hz at node Ⅳ，（e） 352 Hz at node Ⅰ

Fig.11 Gyroscopic metamaterial with defects and displacement fields with two types of defects. （a） GPC with defects on the interface，（b） 451 Hz with common defect，（c） 451 Hz with reverse defect，（d） 352 Hz with common defect，（e） 352 Hz with reverse defect，（f） 102 Hz with common defect，（g） 102 Hz with reverse defect

Fig.12 Gyroscopic metamaterial with line defects and displacement fields with different frequencies

Fig.13 Gyroscopic metamaterial with plane defects and directional displacement fields at the lower band gap

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