Observation of Higher-Order Nodal-Line Semimetal in Phononic Crystals

Higher-order topological insulators and semimetals, which generalize the conventional bulk-boundary correspondence, have attracted extensive research interest. Among them, higher-order Weyl semimetals feature twofold linear crossing points in three-dimensional momentum space, 2D Fermi-arc surface states, and 1D hinge states. Higher-order nodal-point semimetals possessing Weyl points or Dirac points have been implemented. However, higher-order nodal-line or nodal-surface semimetals remain to be further explored in experiments in spite of many previous theoretical efforts. In this work, we realize a second-order nodal-line semimetal in 3D phononic crystals. The bulk nodal lines, 2D drumhead surface states guaranteed by Zak phases, and 1D flat hinge states attributed to $k_z$-dependent quadrupole moments are observed in simulations and experiments. Our findings of nondispersive surface and hinge states may promote applications in acoustic sensing and energy harvesting.

Figure 1

Second-order NLSM for a simple cubic lattice model. (a) Schematic of tight-binding model. The unit cell consists of four sites A–D. Brown cylinders represent the hopping $t_0$, and green cylinders denote the hopping $t_1$. (b) The first BZ and corresponding surface BZ (blue plane). The red dotted lines label the nodal line. (c) Bulk band dispersions along the high-symmetry lines. (d) Upper panel: Zak phase distribution in the $k_x\text{−}{k}_y$ plane. The green (gray) region corresponds to the $\pi$ (0) Zak phase. Lower panel: projected surface dispersions. The green plane denotes the drumhead surface states and the gray area labels the bulk states. (e) Quadrupole moment distribution for the lowest two bulk bands as a function of $k_z$. (f) Projected hinge dispersions along the $k_z$ direction. The cyan lines represent the hinge states. Gray regions and green grids denote the projected bulk and surface states, respectively. The parameters are chosen as $t_1=-0.5$ and $t_0=-1$.

Figure 2

Acoustic second-order NLSM and observation of the bulk nodal lines. (a) Photograph of the fabricated sample for bulk and surface measurements, with an enlarged top view of PC in the inset. (b) Unit cell of the PC. All the colored zones depict the area of air. Acoustic resonators (magenta cuboid) are connected by oblique coupling tubes (green and brown columns). (c) Experimental setup for measuring projected bulk dispersions. The measured pressure field $p$ within the sample at 6.07 kHz is depicted. (d) Bulk dispersions of PC along the high-symmetry lines. The red lines represent the nodal-line degeneracy. (e),(f) Measured and simulated projected bulk dispersions along the $k_x$ direction for several $k_y$.

Figure 3

Experimental observation of the drumhead surface states. (a) Experimental setup for measuring drumhead surface states. The measured pressure field $p$ on surface I at 6.10 kHz is represented. (b) Measured isofrequency contours (left-hand panels) by 2D Fourier transforming the surface fields for different frequencies, compared with the simulated results (right-hand panels). The color maps, green shadows, and gray curves represent the measured results, simulated drumhead surface states, and bulk states, respectively. (c) Measured and simulated projected surface dispersions along the $k_x$ direction for $k_y=0$. (d) Measured and simulated surface dispersions along the $k_y$ direction for $k_x=\pi /a$. In (c) and (d), color maps denote the measured results, and green (gray) lines represent the simulated drumhead surface (projected bulk) states.

Figure 4

Experimental observation of the hinge states. (a) Photograph of the fabricated sample for the hinge state measurement. (b) Measured pressure fields on two surfaces aside from a hinge along the $k_z$ direction at 6.10 kHz. The horizontal directions stand for the real space, while the vertical direction denotes the $k_z$ direction. (c) Measured and simulated projected hinge dispersions along the $k_z$ direction. Cyan (gray) lines represent simulated hinge (projected bulk) states. (d) Measured and simulated projected bulk dispersions along the $k_z$ direction. In (c) and (d), color maps represent measured results, and the cyan (gray) lines denote the simulated hinge (projected bulk) dispersions. The insets give the schematics of the measurement, where the blue circle labels the projections of the source and detector on the $x\text{−}y$ plane.