Abstract
The recent discoveries of higherorder topological insulators (HOTIs) have shifted the paradigm of topological materials, previously limited to topological states at boundaries of materials, to include topological states at boundaries of boundaries, such as corners. So far, all HOTI realisations have been based on static systems described by timeinvariant Hamiltonians, without considering the timevariant situation. There is growing interest in Floquet systems, in which timeperiodic driving can induce unconventional phenomena such as Floquet topological phases and time crystals. Recent theories have attempted to combine Floquet engineering and HOTIs, but there has been no experimental realisation so far. Here we report on the experimental demonstration of a twodimensional (2D) Floquet HOTI in a threedimensional (3D) acoustic lattice, with modulation along a spatial axis serving as an effective timedependent drive. Acoustic measurements reveal Floquet corner states with double the period of the underlying drive; these oscillations are robust, like time crystal modes, except that the robustness arises from topological protection. This shows that spacetime dynamics can induce anomalous higherorder topological phases unique to Floquet systems.
Similar content being viewed by others
Introduction
HOTIs are a class of recently discovered topological phases of matter that extend the standard framework of band topology^{1,2,3,4,5,6,7,8,9,10}. For example, a 2D secondorder topological insulator—unlike a 2D firstorder topological insulator that supports onedimensional (1D) topological edge states—hosts zerodimensional corner states determined by nontrivial higherorder bulk topology (Fig. 1a). This generalised bulkboundary correspondence predicts the existence of topological states at lowerdimensional boundaries (e.g., corners), allowing for the topological characterisation of many materials that would previously have been considered trivial, such as twisted bilayer graphene^{11}. HOTIs have attracted great interest among fields ranging from condensed matter to photonics and acoustics. In particular, they have been realised in various classical ‘metamaterial’ systems^{3,4,5,6,8,9,10}, aided by the ease with which metamaterial properties can be tuned. These previously realised HOTIs have nontrivial structure only in spatial dimensions, limiting the 2D higherorder topology to two major classes characterised by a nontrivial quadrupole moment and a nontrivial polarisation, respectively.
Time is another dimension that can be used to generate interesting bandstructure features. Floquet systems have timeperiodic Hamiltonians satisfying H(t + T) = H(t), where T is a driving period. Such systems can exhibit unconventional topological phases such as Floquet topological insulators^{12,13,14}, which have properties that do not exist in their static counterparts. For example, chiral edge states can exist in an anomalous Floquet topological insulator whose bulk bands all have zero Chern number^{13}, violating the standard bulkedge correspondence principle. There have been many proposals to combine Floquet engineering with higherorder topology^{15,16,17,18,19,20,21,22,23}, but none has been realised previously. Recently, a HOTI has been implemented in a coupledresonator lattice^{24}, which can be described as a Floquet system using a scattering matrix framework^{25,26}; however, the topological phase implemented in that work was a quadrupole insulator, which belongs to the class of static HOTIs.
It has been proposed that Floquet topological states can be utilised to construct perioddoubled oscillations that effectively break discrete timetranslation symmetry. These oscillations are reminiscent of time crystals, but have different origins^{27}: whereas, a time crystal requires quantum manybody interactions to stabilise the breaking of discrete timetranslation symmetry^{28,29}, the perioddoubled Floquet corner states are topologically protected by the spacetime symmetries of the lattice. The observation of this phenomenon, in the original proposal^{27}, requires the coexistence of two distinct Floquet topological phases in a 1D system. The underlying topology, being first order rather than higher order, is also anomalous since it cannot be described with conventional 1D topological invariants.
Here, we experimentally demonstrate an acoustic Floquet HOTI exhibiting topological corner states protected by spacetime symmetries (Fig. 1b). The timedependent drive is simulated by periodic modulation along a spatial axis in a static 3D lattice, which is a common method of realising Floquet dynamics^{14,30}. Unlike static HOTIs, the corner states in the Floquet HOTI can oscillate in time, with oscillation period either equal to, or double of, the driving period. This extends the concept of anomalous Floquet band topology, previously limited to first order, to higher order. The quasienergy bands have zero quadrupole moment in addition to zero polarisation; in a static system, such features would point to a topologically trivial phase. This anomalous Floquet higherorder topology allows for the coexistence of two distinct Floquet topological states, thus satisfying the condition to observe perioddoubled oscillation. We further demonstrate the coexistence of Floquet corner states and Floquet chiral edge states, which constitutes a situation of hybrid topological protection. We perform a series of experiments to probe the various unusual dynamical properties that arise from the system’s Floquet higherorder topology.
Results
Model
There have been many theoretical models proposed to realise Floquet HOTIs^{15,16,17,18,19,20,21,22,23}. Here, we adopt a simple tightbinding model consisting of a 2D bipartite lattice^{23} whose timeperiodic driving protocol is illustrated in Fig. 2a. The model preserves particlehole symmetry and inversion symmetry. The driving protocol consists of four steps with equal duration T/4. In each step, each site only couples to one of its four neighbouring sites (i.e., the instantaneous system is dimerised). A global dimerisation is introduced by letting the coupling strength in one of the four steps differ from other three: the coupling strength is γ for steps 1, 2 and 4, while for step 3 the coupling strength is denoted by θ. By varying γ and θ, we obtain a phase diagram containing various topological phases (see Supplementary Information). The system can exhibit two bandgaps, near quasienergies zero and π. We call these the ‘zero bandgap’ and ‘π bandgap’, and the corner states in these bandgaps ‘zero modes’ and ‘π modes’, respectively. All the quasienergy bandgaps that accommodate corner states are associated with zero quadrupole moment and zero polarisation, which is fundamentally different from previously studied static HOTIs (see Supplementary Information for a discussion of the topological characterisation).
Floquet HOTI with π corner modes
We first consider the case θ = γ = 0.841π. The quasienergy spectrum is plotted in Fig. 2b. There is a π bandgap, which can host corner states, whereas the zero bandgap is closed. To realise this 2D Floquet model, we implement a 3D acoustic lattice (Fig. 2c) with one axis (z) playing the role of time^{14,30}. In all the following calculations and demonstrations, we choose the lattice constant \(a=20\sqrt{2}\ {{{{{{{\rm{mm}}}}}}}}\), and four unit cells extend along both the x and y directions. Each site in the tightbinding model corresponds to a square airhole waveguide with side length l = 10 mm surrounded by hard acoustic boundaries. The coupling between two adjacent sites is accomplished by placing, between two adjacent square waveguides, a few thin connecting channels (these are not visible in Fig. 2c; see Supplementary Information for the design). By modulating the placement of the connecting channels along z, we realise an effective timeperiodic driving following the protocol in Fig. 2a. In this construction, we take L = 336 mm as the modulation period along z. The sample in Fig. 2c has length 3.5L. The coupling strength can be adjusted by altering the number of thin connecting channels; for example, by setting 12 connecting channels, θ = γ = 0.841π can be satisfied at 8000 Hz (see Supplementary Information for the numerical determination of the coupling strengths).
To demonstrate the dynamical properties of the π corner modes, a speaker is placed at z = 0 on the lowerleft corner (indicated by a green arrow in Fig. 2d). The acoustic pressure at different propagation distances is recorded by a microphone (see “Methods”). Figure 2d shows the measured evolution of the corner states, revealing strong localisation around the lowerleft corner. The intensity oscillates between the two sublattices near the corner, each taking half a period, which is a characteristic feature of π modes. These experimental observations are consistent with simulation results (see “Methods” and Supplementary Information), thus verifying the existence of the π modes. Although we have only presented results for frequency 8000 Hz, these dynamical properties of the corner states are observed in a broad frequency range from 7500–8300 Hz (see Supplementary Information).
Anomalous Floquet HOTI with 0 and π corner modes simultaneously
To further explore the properties of the lattice, we take different coupling strengths γ and θ. We consider the effects of reducing the number of connecting channels in step 3 from 12 to 8, such that the coupling strength θ is reduced correspondingly to 0.568π while γ = 0.841π is maintained at 8000 Hz. The resulting quasienergy spectrum is shown in Fig. 3a. In this case, both the π bandgap and zero bandgap are open. The numerically obtained eigenmode profiles in Fig. 3b, c confirm the existence of zero modes and π modes localised at the corners. The π modes oscillate between two sublattices, consistent with Fig. 2d. Despite moderate changes over time, the zero modes mainly concentrate in one sublattice, similar to corner states in static HOTIs.
We then fabricated another experimental sample meeting the condition described in the previous paragraph (θ = 0.568π and γ = 0.841π). The resulting dynamics, shown in Fig. 3d, is very different from the previously studied case that had only π modes present. Under corner excitation, the acoustic intensity is localised around the corner, but the mode profile does not repeat itself after one driving period, as is evident by comparing the acoustic intensities at z/L = 0 and z/L = 1. Instead, a doubled period is observed by comparing acoustic intensities at z/L = 0 and z/L = 2. The period doubling comes from the superposition of zero modes (\(\left0\right\rangle\)) and π modes (\(\left\pi \right\rangle\))—e.g., \(a\left0\right\rangle +b\left\pi \right\rangle\), which evolves to another state after one driving period \({U}_{L}(a\left0\right\rangle +b\left\pi \right\rangle )=(a\left0\right\rangle b\left\pi \right\rangle )\) and comes back to itself at two periods \({U}_{2L}(a\left0\right\rangle +b\left\pi \right\rangle )=(a\left0\right\rangle +b\left\pi \right\rangle )\), where U_{L} is the evolution operator over one period. This period doubling feature has previously been predicted for the breaking of discrete timetranslation symmetry with topological states^{27}, and is a striking outcome of the coexistence of zero and π modes and the anomalous Floquet higherorder topology.
Hybrid Floquet HOTI with 0 corner modes and π chiral edge modes
The zero and π modes that we have observed are protected by the higherorder topology of the zero bandgap and π bandgap, respectively. The topological properties of these two bandgaps can be separately controlled by tuning the coupling strengths γ and θ. Interestingly, it is possible for one bandgap to have nontrivial firstorder topology, while the other exhibits higherorder topology. In that case, two types of protected boundary states—corner states and chiral edge states—can simultaneously exist. To study this phenomenon, we reduce the number of connecting channels in driving steps 1, 2 and 4 to 10 (so that γ = 0.705π) and that in driving step 3 (so that θ = 0.283π). The numerically obtained quasienergy spectrum in Fig. 4a shows that the zero bandgap now hosts zero corner modes (due to higherorder topology), while the π bandgap is spanned by gapless chiral edge states (due to firstorder topology). The calculated eigenmode profiles in Fig. 4b, c confirm that these are indeed coexisting corner states and chiral edge states, subject to their respective topological protection.
We fabricated a sample that meets these conditions (γ = 0.705π and θ = 0.283π). To probe the corner states and edge states separately, we conducted two measurements with different excitations, whose results are plotted in Fig. 4d, e. In the first measurement (Fig. 4d), the source is placed at the lowerleft corner (indicated by the green arrow) at z/L = 0. In this case, the corner state is excited and the acoustic intensity is found to be localised at the corner after an evolution of 3.5 driving periods. In the second measurement (Fig. 4e), the excitation occurs along the left edge (indicated by the green arrow) at z/L = 0. The chiral edge state then propagates along the edge unidirectionally and moves up by around two lattice constants after an evolution of 3.5 driving periods. These observations provide direct evidence of the coexistence of Floquet corner states and Floquet chiral edge states, induced by Floquet engineering.
Discussion
We have proposed and experimentally demonstrated a Floquet HOTI in an acoustic lattice. The Floquet higherorder topology is tied to inversion symmetry and particlehole symmetry (see Supplementary Information), and gives rise to unusual dynamical properties not found in static HOTIs. The coexistence of zero modes and π modes is reminiscent of an edgestatebased time crystal, except that the robustness in this case is tied to topological protection instead of manybody interactions^{27}. These Floquet modes, if reproduced in a quantum lattice, may also find applications in measurementbased quantum computing^{31}. The coexistence of Floquet corner states and Floquet chiral edge states may be useful for state transfer^{32}. These results extend the concept of anomalous Floquet band topology from first order to higher order, substantially expanding the scope of higherorder topological phases (which, in 2D, had been limited to nontrivial quadrupole moment or polarisation). Although the concept has been demonstrated on an acoustic platform, similar models can also be realised in photonic systems such as coupled ring resonators^{24,26} and laserwritten optical waveguides^{14} where the effects of nonHermiticity and nonlinearity can be more easily studied, or even in real timedependent systems^{33}.
Methods
Numerical simulation
The quasienergy spectra in the main text (i.e., Figs. 2b, 3a–c and 4a–c) are obtained from tightbinding calculations, using coupling parameters extracted from Comsol simulations. The procedure for retrieving the coupling parameters is detailed in Sec. VII of the Supplementary Information. The validity of these spectra is verified by fullwave simulations of 3D acoustic structures with periodic boundary conditions along z. At a fixed frequency (8000 Hz in the demonstration), the quasienergy (k_{z}) band can be obtained by restricting our attentions to the modes propagating in the +z direction. The numerical field distributions given in Fig. S13 and Fig. S14 in the Supplementary Information are obtained from Comsol Multiphysics (pressure acoustic module). The boundaries of the 3D printing materials (photosensitive resin) are modelled as rigid acoustic walls due to the large impedance mismatch with air (density ρ = 1.29 kg/m^{3} and sound speed v = 343 m/s). In all fullwave simulations, the models have the same size as the ones used in experiment (z/L = 3.5). The air boundaries at z/L = 0 and z/L = 3.5 are set to be radiation boundaries with an incident field applied to the lowerleft corner or one site on the left edge at z/L = 0.
Sample fabrication
All samples are fabricated through a stereolithography apparatus with ≈ 0.1 mm resolution. To measure the acoustic intensity at different positions along z, each sample is divided into six pieces with cutting positions z/L ∈ {0, 0.5, 1, 1.5, …, 3.5}. These small pieces are fabricated separately and then assembled into the experimental sample.
Experimental measurement
All experiments are conducted using a similar scheme. An acoustic wave is generated by a loudspeaker and guided into one lattice site at z/L = 0 through a small tube. The output signals are recorded by a microphone (Brüel&Kjaer Type 4182) that sweeps all the sites at the output plane. The measured signals are processed by an analyser system (Brüel&Kjaer 3160A022 module) to obtain the frequencyresolved spectrum. In all figures showing experimental results, the data are normalised to the maximal value in the plot.
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
Data availability
The experimental data are available in the data repository for Nanyang Technological University at https://doi.org/10.21979/N9/YBSECE. Other data that support the findings of this study are available from the corresponding authors on reasonable request.
Code availability
All numerical codes are available from the corresponding authors on reasonable request.
References
Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017).
Schindler, F. et al. Higherorder topological insulators. Sci. Adv. 4, eaat0346 (2018).
SerraGarcia, M. et al. Observation of a phononic quadrupole topological insulator. Nature 555, 342–345 (2018).
Peterson, C. W., Benalcazar, W. A., Hughes, T. L. & Bahl, G. A quantized microwave quadrupole insulator with topologically protected corner states. Nature 555, 346–350 (2018).
Imhof, S. et al. Topolectricalcircuit realization of topological corner modes. Nat. Phys. 14, 925–929 (2018).
Noh, J. et al. Topological protection of photonic midgap defect modes. Nat. Photon. 12, 408–415 (2018).
Schindler, F. et al. Higherorder topology in bismuth. Nat. Phys. 14, 918–924 (2018).
Xue, H., Yang, Y., Gao, F., Chong, Y. & Zhang, B. Acoustic higherorder topological insulator on a kagome lattice. Nat. Mater. 18, 108–112 (2019).
Ni, X., Weiner, M., Alu, A. & Khanikaev, A. B. Observation of higherorder topological acoustic states protected by generalized chiral symmetry. Nat. Mater. 18, 113–120 (2019).
Zhang, X. et al. Secondorder topology and multidimensional topological transitions in sonic crystals. Nat. Phys. 15, 582–588 (2019).
Park, M. J., Kim, Y., Cho, G. Y. & Lee, S. B. Higherorder topological insulator in twisted bilayer graphene. Phys. Rev. Lett. 123, 216803 (2019).
Rudner, M. S. & Lindner, N. H. Band structure engineering and nonequilibrium dynamics in Floquet topological insulators. Nat. Rev. Phys. 2, 229–244 (2020).
Rudner, M. S., Lindner, N. H., Berg, E. & Levin, M. Anomalous edge states and the bulkedge correspondence for periodically driven twodimensional systems. Phys. Rev. X 3, 31005 (2013).
Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).
Bomantara, R. W., Zhou, L., Pan, J. & Gong, J. Coupledwire construction of static and Floquet secondorder topological insulators. Phys. Rev. B 99, 45441 (2019).
RodriguezVega, M., Kumar, A. & Seradjeh, B. Higherorder Floquet topological phases with corner and bulk bound states. Phys. Rev. B 100, 85138 (2019).
Seshadri, R., Dutta, A. & Sen, D. Generating a secondorder topological insulator with multiple corner states by periodic driving. Phys. Rev. B 100, 115403 (2019).
Nag, T., Juričić, V. & Roy, B. Out of equilibrium higherorder topological insulator: Floquet engineering and quench dynamics. Phys. Rev. Res. 1, 32045 (2019).
Peng, Y. & Refael, G. Floquet secondorder topological insulators from nonsymmorphic spacetime symmetries. Phys. Rev. Lett. 123, 16806 (2019).
Ghosh, A. K., Paul, G. C. & Saha, A. Higher order topological insulator via periodic driving. Phys. Rev. B 101, 235403 (2020).
Hu, H., Huang, B., Zhao, E. & Liu, W. V. Dynamical singularities of Floquet higherorder topological insulators. Phys. Rev. Lett. 124, 57001 (2020).
Huang, B. & Liu, W. V. Floquet higherorder topological insulators with anomalous dynamical polarization. Phys. Rev. Lett. 124, 216601 (2020).
Zhu, W., Chong, Y. D. & Gong, J. Floquet higherorder topological insulator in a periodically driven bipartite lattice. Phys. Rev. B 103, L041402 (2021).
Mittal, S. et al. Photonic quadrupole topological phases. Nat. Photon. 13, 692–696 (2019).
Pasek, M. & Chong, Y. D. Network models of photonic Floquet topological insulators. Phys. Rev. B 89, 075113 (2014).
Gao, F. et al. Probing topological protection using a designer surface plasmon structure. Nat. Commun. 7, 11619 (2016).
Bomantara, R. W. & Gong, J. Simulation of nonAbelian braiding in Majorana time crystals. Phys. Rev. Lett. 120, 230405 (2018).
Zhang, J. et al. Observation of a discrete time crystal. Nature 543, 217–220 (2017).
Choi, S. et al. Observation of discrete timecrystalline order in a disordered dipolar manybody system. Nature 543, 221–225 (2017).
Peng, YuGui et al. Chiralityassisted threedimensional acoustic Floquet lattices. Phys. Rev. Res. 1, 33149 (2019).
Bomantara, R. W. & Gong, J. Measurementonly quantum computation with Floquet Majorana corner modes. Phys. Rev. B 101, 085401 (2020).
Bomantara, R. W. Timeinduced secondorder topological superconductors. Phys. Rev. Res. 2, 033495 (2020).
Darabi, A., Ni, X., Leamy, M. & Alù, A. Reconfigurable Floquet elastodynamic topological insulator based on synthetic angular momentum bias. Sci. Adv. 6, eaba8656 (2020).
Acknowledgements
B.Z. and Y.C. acknowledge funding support by the Singapore Ministry of Education Academic Research Fund Tier3 (Grant No. MOE2016T31006) and Tier 2 (Grant No. MOE2019T22085). J.G. acknowledges funding support by the Singapore Ministry of Education Academic Research Fund Tier3 Grant No. MOE2017T31001 (WBS No. R144000425592) and by the Singapore NRF Grant No. NRFNRFI201704 (WBS No. R144000378 281). We are grateful to H.X. Sun, R. Bomantara and S. Mu for helpful discussions.
Author information
Authors and Affiliations
Contributions
W.Z. and H.X. contributed equally to this work. W.Z. and H.X. carried out the simulation. H.X. and W.Z. designed and performed the experiment. J.G., Y.C. and B.Z. supervised the project. All authors contributed extensively to the proposal of idea, the interpretation of the results and the writing of the paper.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review information
Nature Communications thanks Meng Xiao and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Zhu, W., Xue, H., Gong, J. et al. Timeperiodic corner states from Floquet higherorder topology. Nat Commun 13, 11 (2022). https://doi.org/10.1038/s41467021275526
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467021275526
This article is cited by

Robust temporal adiabatic passage with perfect frequency conversion between detuned acoustic cavities
Nature Communications (2024)

Fractal photonic anomalous Floquet topological insulators to generate multiple quantum chiral edge states
Light: Science & Applications (2023)

Topological invariants for anomalous Floquet higherorder topological insulators
Frontiers of Physics (2023)

Topological acoustics
Nature Reviews Materials (2022)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.