Higher-order Klein bottle topological insulator in three-dimensional acoustic crystals

Topological phases of matter are classified based on symmetries, with nonsymmorphic symmetries like glide reflections and screw rotations being of particular importance in the classification. In contrast with extensively studied glide reflections in real space, introducing space-dependent gauge transformations can lead to momentum-space glide reflection symmetries, which may even change the fundamental domain for topological classifications, e.g., from a torus to a Klein bottle. Here, we discover a class of three-dimensional (3D) higher-order topological insulators, protected by a pair of momentum-space glide reflections. It supports gapless hinge modes, as dictated by the quadrupole moment and Wannier Hamiltonians defined on a Klein bottle manifold, and we introduce two topological invariants to characterize this phase. Our predicted topological hinge modes are experimentally verified in a 3D-printed acoustic crystal, providing direct evidence for 3D higher-order Klein bottle topological insulators. Our results not only showcase the remarkable role of momentum-space glide reflections in topological classifications but also pave the way for experimentally exploring physical effects arising from momentum-space nonsymmorphic symmetries.

Figure 1

(a) Schematic illustration of a tight-binding model for the HOKBTI. The red bonds indicate the hopping with the phase of −1 compared with the hopping through blue bonds. (b) and (c) Schematics of reflection symmetries with respect to (b) $x$-normal and (c) $y$-normal planes. In (b), the lattice is reflected about the red plane by $M_x$ followed by applying a gauge transformation $[G_{M_x}=(-1)z]$. The gauge transformation is described by imposing the phases in the brackets on the corresponding hopping. It follows that the combined operation ${\mathcal{M}}_{\mathrm{x}}=G_{M_x}{M}_x$ does not change the hopping configuration, and thus, the Hamiltonian remains invariant through this operation. In (c), $G_{M_y}={\left(-1\right)}^{z+1}{\tau }_3{\sigma }_3$.

Figure 2

(a) Momentum-space glide reflection symmetry, $m_x\mathcal{H}\left(\mathbf{k}\right)m_x^{-1}=\mathcal{H}\left(-k_x,k_y,k_z+\pi \right)$ (top) and $m_y\mathcal{H}$(k) $m_y^{-1}=\mathcal{H}\left(k_x,-k_y,k_z+\pi \right)$ (bottom), resulting in the Brillouin Klein bottle at each slice with a fixed $k_y$ (top) or $k_x$ (bottom). (b) Schematic illustration of how the Klein bottle base space arises for the Wannier Hamiltonian. (c) The quadrupole moment $q_{xy}\left(k_z\right)$ vs $k_z$ for the Hamiltonian in Eq. (3) with $t_x=t_y=0.1$ (blue line) and $t_x=t_y=1.1$ (red line). (d) Energy spectra vs $k_z$ for the Hamiltonian in Eq. (3) at $t_x=t_y=0.1$ with open and periodic boundary conditions in the $x-y$ plane and along $z$, respectively. The blue and red lines represent the hinge modes localized at diagonal hinges (solid blue circles) and off-diagonal hinges (solid red circles), respectively. In (c) and (d), $t{{}_x}^{\prime }=t{{}_y}^{\prime }=1$.

Figure 3

Simulated results of the positive and negative couplings. (a) The dipole mode supported in a single cavity. (b)–(e) Acoustic pressure field distributions for different linking structures. One can also find the resonant frequency and type of the dipole mode beside the structure.

Figure 4

(a) A photo of the three-dimensional (3D)-printed acoustic crystal sample. The four green stars show the positions of four acoustic sources labeled ${\mathrm{S}}_1\text{–}{\mathrm{S}}_4$. (b) Schematic illustration of a unit cell for the acoustic crystal. The blue and red rectangular tubes between resonators represent the couplings with positive and negative signs, respectively.

Figure 5

(a) and (b) Measured acoustic dispersions (colored) of the hinge states with positive or negative group velocities. The gray dots represent the simulated bulk and hinge dispersions. (c) Simulated (left) and measured (right) acoustic pressure field distributions of the hinge states at the frequency of 2.5 kHz. In (c) and (d), the color bars are normalized to the maxima so that the value changes from 0 to 1.

Figure 6

Measured acoustic pressure field distributions at the frequency of (a) 2.34 kHz and (b) 2.68 kHz.

Figure 7

(a) Loss of the acoustic waves along the $z$ direction. Red triangles and black squares denote the experimental and simulated results (which are normalized by their maximum values), respectively. Their fitting curves (solid lines) have almost the same slopes, indicating consistent attenuation behavior. (b) The measured transmission spectrum of hinge (blue) and bulk (black) states by putting the source at the corner and center of bottom surface and detector at the corner and center of the top surface, respectively. The gray region highlights the bulk band gap.

Figure 8

Quadrupole moments, energy gaps, and energy spectra for the Hamiltonian in Eq. (3) with the term $H_{\mathrm{\Delta }}=\mathrm{\Delta }{\tau }_0{\sigma }_3$. (a) The quadrupole moment $q_{xy}$ ($k_z$) vs $k_z$ for different Δ. (b) The bulk energy gap (blue line), $x$-normal surface energy gap (red line) and $y$-normal surface energy gap (green line) vs Δ. The energy spectrum with open boundary conditions along $x$ and $y$ and periodic boundary conditions along $z$ for (c) $\mathrm{\Delta }=0$ and (d) $\mathrm{\Delta }=1.2$, respectively. We also set $a_x=a_y=a_z=1$.

Figure 9

Energy spectra, quadrupole moment, and polarizations for the model with long-range hoppings. (a) Then energy spectrum with respect to $k_z$ for a system with open boundary conditions along $x$ and $y$ and periodic boundary conditions along $z$. The blue line represents the hinge modes localized at the upper left and lower right hinges, and the red line represents the hinge modes localized at the upper right and lower left hinges [also see Fig. 2]. (b) The quadrupole moment $q_{xy}$ as a function of $k_z$. (c) The Wannier-sector polarizations $p_x$ (red line) and $p_y$ (blue line) with respect to $k_z$. The insets are the zoomed-in view in the interval $k_z\in [0.475\pi ,0.485\pi ]$. Here, $t_x=0.3$, $t_y=0.6$, $t{{}_x}^{\prime }=t{{}_y}^{\prime }=1$, $t_z=0.5$, $t_1=0.1$, $t_2=0.4$, and $t_3=0.4$. We also set $a_x=a_y=a_z=1$.

Figure 10

(a) The phase diagram of $\chi$ and ${\chi }_q$ as a function of $\gamma$ for the Hamiltonian in Eq. (D2). In the gray regions, $\chi$ and ${\chi }_q$ are not equal. (b) The energy spectrum vs $\gamma$ for the system under open boundary conditions along $x$ and $y$ and periodic boundary conditions along $z$. The energy spectrum vs $k_z$ at (c) $\gamma =-0.75$ and (d) $\gamma =0.05$. Here, $b_2=1.2$, $g_0=0.65$, $t_x=1$, $t_y=1$, and $t_z=0.5$.

Figure 11

Demonstration of positive and negative couplings between two acoustic cavities. (a) and (b) Negative and positive couplings in the $x-y$ plane. (c) and (d) Negative and positive couplings along the $z$ direction. Upper panels: Pictures of the two coupled acoustic cavities and experimental setups. An acoustic point source is placed on the right side of a cavity to excite an acoustic pressure field, and the pressure responses are detected at four typical positions marked as A–D. The two cavities are coupled by a narrow tube with different geometric configurations to realize either positive or negative couplings. Middle panels: Measured amplitude responses at four positions A (black), B (red), C (blue), and D (green). The gray arrows point out the numerically calculated resonant frequencies. Lower panels: Corresponding measured phase spectra.

Figure 12

(a) Measured and (b) simulated pressure field distributions of the hinge states in the $x\text{−}y$ plane in Fig. 4 at the frequency of 2.5 kHz. The fields are normalized to their maxima.

Figure 13

(a) The energy spectrum vs $k_z$ for the Hamiltonian in Eq. (3) including the on-site disorder term $H_{\mathrm{d}}$ under open boundary conditions in the $x\text{−}y$ plane. Here, we only consider one random configuration with $W=0.05$. The blue and red lines denote the hinge modes localized at diagonal and off-diagonal hinges, respectively. (b) The configuration averaged $\overline{{\chi }_q}$ as a function of $W$. $\overline{{\chi }_q}$ suddenly drops from 1 at $W\approx 1.46$ (dotted line), indicating the occurrence of a phase transition. (c) The configuration averaged local density of state $\overline{\rho \left(E,\mathit{r}\right)}$ at zero energy for disordered systems with $W=0.05$ under open boundary conditions in the $x\text{−}y$ plane. The distribution of disorder respects two reflection symmetries. Here, $t_x=t_y=0.1$, and the size in the $x\text{−}y$ plane is $10\times 10$. The number of random configurations in (b) and (c) is 100. We also set $a_x=a_y=a_z=1$.

Figure 14

(a) The phase diagram as a function of $t_x$ and $t_y$ for the Hamiltonian in Eq. (G1). The light red region represents the topologically nontrivial phase with ${\chi }_q=1$ hosting gapless hinge modes in the energy spectrum. The dark gray regions depict the topologically nontrivial phase with ${\chi }_q=1$ supporting gapless hinge modes only in the entanglement spectrum. (b) Schematics for calculating the entanglement spectrum in the quarter subsystem $A$ in the $x\text{−}y$ plane. (c) The energy spectrum and (d) the entanglement spectrum with respect to $k_z$ for the Hamiltonian with $t_x=1.2$ and $t_y=0.1$ [marked as a solid yellow circle in (a)].