Acoustic Möbius Insulators from Projective Symmetry

In the presence of gauge symmetry, common but not limited to artificial crystals, the algebraic structure of crystalline symmetries needs to be projectively represented, giving rise to unprecedented topological physics. Here, we demonstrate this novel idea by exploiting a projective translation symmetry and constructing a variety of Möbius-twisted topological phases. Experimentally, we realize two Möbius insulators in acoustic crystals for the first time: a two-dimensional one of first-order band topology and a three-dimensional one of higher-order band topology. We observe unambiguously the peculiar Möbius edge and hinge states via real-space visualization of their localiztions, momentum-space spectroscopy of their $4\pi$ periodicity, and phase-space winding of their projective translation eigenvalues. Not only does our work open a new avenue for artificial systems under the interplay between gauge and crystalline symmetries, but it also initializes a new framework for topological physics from projective symmetry.

Figure 1

2D first-order Möbius insulator and 3D higher-order Möbius insulator from projective symmetry. (a) Unit cell of the 2D MI (left) and its effective decomposition (right). (b) Bulk band structures calculated with hoppings $t=1$ and ${\gamma }_x={\lambda }_x=2$ (magenta dashed lines) and $t={\gamma }_x=1$ and ${\lambda }_x=4$ (black solid lines). Each band is twofold degenerate. (c) Edge-projected band structure in the $z$ direction for the gapped case in (b) featuring a Möbius twist. (d) Unit cell of the 3D HOMI (left) and its effective decomposition (right). (e) Bulk band structure exemplified by a HOMI with hoppings $t={\gamma }_x={\gamma }_y=1$ and ${\lambda }_x={\lambda }_y=4$. Each band is fourfold degenerate. (f) The hinge-projected band structure in the $z$ direction for the case in (e). The sparser black lines are projected surface states. In (c) and (f), the Möbius twist is formed by two $\pi$-crossed, $4\pi$-periodic, bulk-decoupled bands of opposite projective translation eigenvalues (${\ell }_{\pm }=\pm e^{i{k}_z/2}$). The color scale indicates the phase profile of ${\ell }_{\pm }$.

Figure 2

3D first-order Möbius insulator and Möbius Dirac semimetal from projective symmetry. (a),(b) Bulk band structures at $k_z=\pi$ exemplified for a 3D MI and a 3D Möbius Dirac semimetal. Each band is twofold degenerate. (c),(d) The corresponding band structures projected into the $k_y\text{−}{k}_z$ surface, featuring Möbius-twisted surface states. In (c), the Möbius twist has a zero-energy line degeneracy at $k_z=\pi$ traversing the surface Brillouin zone. In (d), the Möbius twist has a similar line degeneracy but only between the two projected Dirac points.

Figure 3

Acoustic realization of a 2D Möbius insulator and edge states. (a) Experimental sample. The magenta circle and star label the positions of the sound source in the bulk and edge measurements, respectively. Inset: the unit-cell geometry of our acoustic crystal, where the air cavities (white) and narrow tubes (color) mimic the orbitals and hoppings in the tight-binding model, respectively. The lattice constants are $a=c=75\mathrm{mm}$. (b) Experimentally measured (color scale) and theoretically predicted (black line) bulk spectra. (c) Left: frequency-resolved pressure amplitude scanned along the dashed line in (a), where the two yellow dashed lines indicate the frequency window predicted for the edge states. Right: the data extracted at 5720 Hz (magenta spheres) plotted in log scale. (d) Intensity profiles at two selected frequencies respectively for the bulk and edge states, measured for a sample of samller size. (e) Measured (color scale) and predicted (black line) edge spectra. (f) Measured phase information of ${\ell }_{\pm }$ (color dots) encoding the measured edge bands, compared with the theoretical results (color lines).

Figure 4

Acoustic realziation of a 3D higher-order Möbius insulator and hinge states. (a) Experimental sample, where the inset sketches the unit-cell geometry of our 3D acoustic crystal. The lattice constants are $a=b=72.6\mathrm{mm}$ and $c=43.6\mathrm{mm}$. (b) Measured phase information of ${\ell }_{\pm }$ (color dots) encoding the measured hinge bands, compared with the theoretical results (color lines). The black lines are projected bulk and surface states. (c)–(e) Intensity profiles measured at three frequencies for the bulk, surface, and hinge states, respectively.