Observation of a phononic quadrupole topological insulator

Abstract

The modern theory of charge polarization in solids1,2 is based on a generalization of Berry’s phase3. The possibility of the quantization of this phase4,5 arising from parallel transport in momentum space is essential to our understanding of systems with topological band structures6,7,8,9,10. Although based on the concept of charge polarization, this same theory can also be used to characterize the Bloch bands of neutral bosonic systems such as photonic11 or phononic crystals12,13. The theory of this quantized polarization has recently been extended from the dipole moment to higher multipole moments14. In particular, a two-dimensional quantized quadrupole insulator is predicted to have gapped yet topological one-dimensional edge modes, which stabilize zero-dimensional in-gap corner states14. However, such a state of matter has not previously been observed experimentally. Here we report measurements of a phononic quadrupole topological insulator. We experimentally characterize the bulk, edge and corner physics of a mechanical metamaterial (a material with tailored mechanical properties) and find the predicted gapped edge and in-gap corner states. We corroborate our findings by comparing the mechanical properties of a topologically non-trivial system to samples in other phases that are predicted by the quadrupole theory. These topological corner states are an important stepping stone to the experimental realization of topologically protected wave guides12,15 in higher dimensions, and thereby open up a new path for the design of metamaterials16,17.

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Figure 1: Quadrupole topological insulator.
Figure 2: Quadrupole in-gap states.
Figure 3: Edge and corner modes.
Figure 4: Reduced model and Wannier bands.

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Acknowledgements

We acknowledge financial support from the Swiss National Science Foundation and the NCCR QSIT. T.L. acknowledges support from a Marie Curie fellowship and O.R.B. from the ETH postdoctoral fellowship FEL-26 15-2.

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Authors

Contributions

S.D.H. conceived the research. M.S.-G., V.P. and O.R.B. designed the samples. M.S.-G., V.P., S.D.H. and R.S. conducted the experiments. L.G.V. and T.L. fabricated the samples.

Corresponding author

Correspondence to Sebastian D. Huber.

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Extended data figures and tables

Extended Data Figure 1 Phase diagram.

a, Phase diagram of the model in equation (3). The brown area marks the quantized quadrupole phase (1/2, 1/2), whereas the orange areas are the (1/2, 0) and (0, 1/2) phases with no corner modes but emergent edge physics along two parallel edges. The dashed line indicates the C4-symmetric line, where the bulk gap is closing at the phase transition. The transitions away from the C4-symmetric line happen through bulk-induced edge transitions, where no bulk gap is closing. b, The evolution of the Wannier bands in the x and y directions along the path shown in a. The transition from the quadrupole phase to the (1/2, 0) phase is marked by a gap closing at 1/2, removing any polarization in the system. The second transition is induced by a gap closing at 0.

Extended Data Figure 2 Transducer characterization.

The sound pressure level (SPL) in the frequency response of the ultrasound transducer that we used over the frequency region of interest (shaded in grey). The 0.46 dB fluctuations are negligible with respect to the 80 dB variations in the measured response.

Extended Data Figure 3 Corner Greens functions.

ad, The corner plates at (0, 9) (a), (9, 9) (b), (0, 0) (c) and (9, 0) (d) are excited at the respective edge-mode frequency. The response recorded (amplitude and phase) enables us to reconstruct the eigenfunctions ψ(x, y) of the individual corner modes. Along each edge, the measured decay of the modes (black) is shown together with the theoretical prediction (orange). Given the decay length ξ/a ≈ 1.6, where a is the lattice constant, the residual weight of at most 2% at the corners other than the one that is excited stems from spurious acoustic excitation rather than hybridization.

Extended Data Figure 4 Edge Greens function.

a, Sketch of the measured system. The bottom-left plate at (x, y) = (0, 0) was excited. b, Integrated frequency response , where i runs along the red and black edges indicted by the solid lines in a. The two highest-frequency peaks below the bandgap (indicated by vertical lines) are analysed. c, The decay of the two edge modes into the bulk (black) along the dashed lines in a. The orange lines show the theoretical predictions. d, The mode profiles ψ(x, y) along the two edges. The edge with positive couplings has nodes between the unit cells (top panel, black), whereas the edge with negative couplings has nodes inside the unit cell (bottom panel, red), which establishes that a direct measurement of the negative couplings gives rise to the π flux.

Extended Data Figure 5 Perturbative design of a quadrupole topological insulator.

a, The design approach is based on establishing a correspondence between elements of an objective model (left) and geometric features of the metamaterial (right). Each degree of freedom of the objective model is mapped into a single plate (yellow arrows) by expressing the displacement of each plate as a linear combination of free-plate modes. b, Here, only the first non-rigid-body mode, which has dxy symmetry, is used (top left). The other modes are the second, third and fourth non-rigid-body modes. c, Independent two-plate systems simulated to create an adequate initial guess for the geometry of the system. d, Four-unit-cell design simulated during the final gradient optimization. e, The refined single-plate design removes material at the maximums of nearby higher-order modes. f, Small trenches at the junction between beams and plates. These trenches suppress the coupling to higher-order modes by avoiding regions where these modes have a large displacement. g, Dispersion along high-symmetry lines in the Brillouin zone calculated using the finite-element method. The bands that arise from the dxy mode are highlighted in colour. h, Detailed view of the spectrum in the frequency range of interest. The dots denote the full finite-element results whereas the lines are calculated from the extracted reduced-order model.

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Serra-Garcia, M., Peri, V., Süsstrunk, R. et al. Observation of a phononic quadrupole topological insulator. Nature 555, 342–345 (2018). https://doi.org/10.1038/nature25156

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