Topological Creation of Acoustic Pseudospin Multipoles in a Flow-Free Symmetry-Broken Metamaterial Lattice

Snowflake phononic topological insulator at the nanoscale

“…So far, experimental implementations exist on the centimeter scale, both for the case of time-reversal symmetry broken by external driving [1], such as in coupled gyroscopes, as well as for the case without driving [2][3][4][5][6], such as in coupled pendula. Moreover, a multitude of different implementations have been envisioned theoretically [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. However, it is highly desirable to come up with alternative design ideas that may be realized on the nanoscale, eventually pushing towards applications in integrated phononics.…”

Experimental verification of acoustic pseudospin multipoles in a symmetry-broken snowflakelike topological insulator

“…By varying the filling fraction such that d = 0.85 cm, a double twofold degeneracy, one for the lower bands of the p type and the other for the upper bands of the d type, appears at the Brillouin zone (BZ) center, as shown in Fig. 1(c) [28]. Interestingly, a pair of dipolar resonance states are accompanied by a pair of quadrupolar resonance states: similar to p and d orbitals of electrons, here the dipoles are even or odd symmetrical to the axes x/y (corresponding to p x /p y ), while the quadrupole is odd symmetrical to the axes x and y (corresponding to d xy ), or even symmetrical to the axes x and y at the same time (corresponding to d x 2 −y 2 ).…”

“…Topological states have also been extended to condensed-matter physics based on the quantum Hall effect (QHE) [1,2], the quantum spin Hall effect (QSHE) [3,4], and topological insulators (TIs) [5,6]. Over the past ten years, investigation into new topologically protected edge states has started to grow in other subfields of physics, such as photonics [7][8][9][10][11][12][13][14][15][16], phononics [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32], and mechanics [33][34][35][36]. The intrinsic difference between electrons and acoustic waves represents a great challenge in creating the spinlike degree of freedom for sound only possessing longitudinal polarization.…”

“…However, to construct an artificial acoustic Kramers doublet, the degree of freedom must be increased to obtain a pair of pseudospin states analogous to electrons [27,28]. As a result, fourfold-degenerate double Dirac cones in acoustic band structure are required, which can be engineered in a triangular/honeycomb lattice [41,42].…”

“…However, the inherent losses and noise that intrinsically accompany acoustic propagation in moving media, together with considerable fabrication complexities of ring waveguides may become detrimental in future topological applications. Most recently, phononic "graphene" with double Dirac cones [27][28][29] has been proposed for the design of two-dimensional (2D) acoustic T -symmetric TIs, functioning without applying external "forces." The band inversion between pseudospin states has been induced by varying the radius of the constitutive meta-atoms.…”