Discovery of Higher-Order Nodal Surface Semimetals

The emergent higher-order topological insulators significantly deepen our understanding of topological physics. Recently, the study has been extended to topological semimetals featuring gapless bulk band nodes. To date, higher-order nodal point and line semimetals have been successfully realized in different physical platforms. However, for the conceptually expected higher-order nodal surface semimetals, a concrete model has yet to be proposed, let alone experimentally observed. Here, we report an ingenious design route for constructing this unprecedented higher-order topological phase. The three-dimensional model, layer-stacked with a two-dimensional anisotropic Su-Schrieffer-Heeger lattice, exhibits appealing hinge arcs connecting the projected nodal surfaces. Experimentally, we realize this new topological phase in an acoustic metamaterial, and present unambiguous evidence for both the bulk nodal structure and hinge arc states, the two key manifestations of the higher-order nodal surface semimetal. Our findings can be extended to other classical systems such as photonic, elastic, and electric circuit systems, and open new possibilities for controlling waves.

Figure 1

Summary of 3D HO topological semimetals. In contrast to the 3D HO TI (a) with 1D hinge states over the projected BZ, the 3D HO NP (b), NL (c), and NS (d) semimetals feature hinge arcs connecting the projected nodal objects.

Figure 2

2D ASSH model. (a) Tight-binding model. The four orbitals coupled with anisotropic hoppings belong to two decoupled sublattices, as distinguished by the balls of different color. Inset: 2D BZ. (b) Phase diagram plotted for fixed $\beta =1$. It includes a gapless NL semimetal phase, in addition to the gapped trivial insulator ($\mathcal{N}=0$) and HO TI ($\mathcal{N}=1$) phases. (c) Band structures for a sequence of systems with fixed $\alpha =3$ but varied $\lambda$ [see the dots in (b)], which exhibit a ringlike 2D NL within $0.5<\lambda <2$. (d) Finite-size spectra for three specified $\lambda$ values. In contrast to the trivial insulator (left) and NL semimetal (middle) phases, the 2D HO TI (right) hosts four nontrivial zero-energy corner states (inset).

Figure 3

Tight-binding model for constructing 3D HO NS semimetals. (a) Schematic of sweeping 2D NLs to form a 3D NS. (b) 3D lattice stacked by the 2D ASSH model. (c) Conceiving 3D topologically distinct phases from the effective hopping ratio ${\lambda }_{\mathrm{eff}}(k_z)=\lambda +2\gamma \cos k_z$. Again, we consider $\alpha =3$ and $\beta =1$. The blue and red circles sketch possible $k_z$-slice evolutions to generate a 3D HO TI and a 3D HO NS, respectively. (d) Visualization of the NS structure for the specified $\lambda$ and $\gamma$. Each rugbylike NS carries a nontrivial ${\mathbb{Z}}_2$ charge $\nu =1$. (e) Bulk band structure along high-symmetry lines. (f) Hinge-projected spectrum, which features zero-energy hinge arcs (red line) connecting the projected NSs. Inset: quadrupole chiral number $\mathcal{N}$ plotted as a function of $k_z$.

Figure 4

Acoustic demonstration of the HO topology in 2D ASSH model. (a) Experimental sample of $10\times 10$ unit cells. Inset: unit cell geometry. (b) Simulated (lines) and measured (color) bulk dispersions. (c) Frequency spectrum simulated for the finite-size sample, which shows four degenerate corner states pinned to the middle of bulk gap. Inset: pressure intensity profile of the corner mode, where the gray line sketches $1/4$ sample. (d) Intensity spectra measured at bulk and corner sites. Inset: intensity pattern extracted for the peak frequency of the corner spectrum, demonstrated with $1/4$ sample for clarity.

Figure 5

Experimental evidence for our 3D acoustic HO NS semimetal. (a) Experimental sample. The red stars label the sound sources positioned for measuring the surface and hinge spectra. (b) Unit cell structure. The yellow and gray tubes specify the interlayer couplings. (c) Measured (color) and simulated (lines) surface spectra for the $XY$ and $XZ$ surfaces. The dashed yellow boxes signify the projected NS structure. (d) Equifrequency map experimentally extracted at 8.60 kHz (color), which capture the simulated NS projections (transparent cyan) to different surface BZs. (e) Hinge spectra measured and simulated along the $k_z$ direction. (f) Pressure distributions measured at 8.60 and 8.30 kHz, clearly showing the hinge (left) and bulk (right) states, respectively.