Experimental Observation of Non-Abelian Earring Nodal Links in Phononic Crystals

Nodal lines are symmetry-protected one-dimensional band degeneracies in momentum space, which can appear in numerous topological configurations such as nodal rings, chains, links, and knots. Very recently, non-Abelian topological physics have been proposed in space-time inversion ($PT$) symmetric systems. One of the most special configurations in such systems is the earring nodal link, composing of a nodal chain linking with an isolated nodal line. Such earring nodal links have not been observed in real systems. We designed phononic crystals with earring nodal links, and experimentally observed two different kinds of earring nodal links by measuring the band structures. We found that the order of the nodal chain and line can be switched after band inversion but their link cannot be severed. Our Letter provides experimental evidence for phenomena unique to non-Abelian band topology and our acoustic system provides a convenient platform for studying the new materials carrying non-Abelian charges.

Figure 1

(a) Nontrivial accidental triple degeneracy in momentum space. The red (blue) nodal lines are formed by the lower (upper) pair of bands. (b),(c) The two earring nodal link configurations when two different perturbations (changing the value of $b$ from 0 to 0.15 and $-0.15$, respectively) are added to the case of the nontrivial accidental triple degeneracy. (d) The dispersion at $k_z=0$ for (a). (e) Eigenstates on the $k_z=0$ plane, where the three color bars represent the three orthogonal eigenstates and the gray circle corresponds to the green loop in (a). (f) The rotation of the eigenstates frame on the unit sphere along green loop in (a), where the spheres with the same color trace out the trajectory of the eigenstates of each band. In (d)–(f), cyan, light magenta, and green colors correspond to bands 1, 2, and 3, respectively.

Figure 2

(a) The unit cell of the phononic crystals. (b)–(d) The corresponding band structures along $k_z$ with $w_1=1.6$, 3.0, and 3.9 mm. $(+,+)$ indicates that the sound pressure field of the eigenmode is even in both the $x$ and $y$ directions at the intersection of the two mirror planes $k_x=k_y=0$. $(+,-)$ indicates that the pressure field is an even mode in the $x$ direction and an odd mode in the $y$ direction, and $(-,+)$ indicates that the pressure field is an odd mode in the $x$ direction and an even mode in the $y$ direction. The black dots mark three eigenstates in bands 2–4 with different symmetries, and the distribution of the pressure fields is shown in the illustration. The red (blue) dots represent the nodal points of band 23 (band 34). (e)–(g) The distribution of nodal lines in (e)–(g) corresponding to the band structures in (b)–(d), respectively. The red and blue dots in (e)–(g) correspond to the red and blue nodal points in Figs. 3 [Figs. 3]. (h) The rotation of the eigenstate frame on the unit sphere along green loop at $k_z=0.5\pi /h$ in full-wave simulations.

Figure 3

(a) Experimental sample-1 [the structural parameters are the same as in Fig. 2]. The red star is the position of the point source. (b)–(d) Experimental (color maps) and theoretical (white lines) equal-frequency contours at $k_y=0$ and $k_x=0$ for different frequencies for sample 1. The theoretical nodal points are marked by red or blue dots. (e) Experimental sample 2 [the structural parameters are the same as in Fig. 2]. (f)–(g) Experimental and theoretical band structures at $k_y=0$ and $k_x=0$ for different frequencies for sample 2. The theoretical nodal points are marked by red or blue dots.