Observation of an acoustic octupole topological insulator

Berry phase associated with energy bands in crystals can lead to quantised observables like quantised dipole polarizations in one-dimensional topological insulators. Recent theories have generalised the concept of quantised dipoles to multipoles, resulting in the discovery of multipole topological insulators which exhibit a hierarchy of multipole topology: a quantised octupole moment in a three-dimensional bulk induces quantised quadrupole moments on its two-dimensional surfaces, which in turn induce quantised dipole moments on one-dimensional hinges. Here, we report on the realisation of an octupole topological insulator in a three-dimensional acoustic metamaterial. We observe zero-dimensional topological corner states, one-dimensional gapped hinge states, two-dimensional gapped surface states, and three-dimensional gapped bulk states, representing the hierarchy of octupole, quadrupole and dipole moments. Conditions for forming a nontrivial octupole moment are demonstrated by comparisons with two different lattice configurations having trivial octupole moments. Our work establishes the multipole topology and its full hierarchy in three-dimensional geometries.

This is an experimental paper in which the authors designed, fabricated and characterized a 3D acoustic metamaterial which is a quantized octupole TI. They observed 0D corner states, 1D hinge states, 2D surface states, and 3D bulk states, representing the topological hierarchy of quantised octupole, quadrupole and dipole moments. This is a nice piece of work. They gave a rigorous treatment for the tight-binding model and they implemented the model with a simple and clever design that achieved negative hopping. They gave control experiments of trivial phases which makes their case more convincing. The presentation is clear and concise. I think the paper amply meets the expectation of Nature Commun.
I only find some rather minor issues which the authors might want to address.
1. According to my understanding, whether the coupling is positive or negative (as illustrated in Fig.1d) depends on the convention of the phase of the dipolar eigenstate in each cavity. There is no reason why I cannot regard the two modes in the right panel as "positive coupling", but the two coupled modes in the left panel as "negative coupling". This new convention is equivalent to the original one up to a gauge transformation. In the full structure, there are 4 cavities in each unit cell, and the cavities have different orientations. The signs of the coupling bonds are not obvious unless all the phases of the dipolar cavity modes are fixed. Therefore, I think the authors should give some explanation about the their convention more explicitly, and clarify that the topology of the system does not depend on the convention of the gauge.
2. Since the fields inside the cavities have inhomogeneous distributions, it is not obvious how the acoustic intensity at each site shown in Fig. 2,3,4 is defined. Do they refer to the field intensity at a certain point in the cavity or to the average intensity inside the whole cavity?
3. The discussion of robustness of the corner states is somewhat misleading. In fact, the statements of robustness of the topological effects have sometimes been used too loosely in the field of classical topological waves. In the present case, it seems that both the two types of perturbations shown in the supplementary Fig. S4 violate the underlying symmetries (the two anti-commutative mirror symmetries) that protect the quantization of the octupole momentum. Symmetry-protected topological phases are only stable against the perturbations preserving the underlying symmetries. If the symmetry constraints are compromised, a system can continuously change from a SPT nontrivial phase to a SPT trivial phase without gap closing. Though the original Science paper, "quantized electric multipole insulators", also claimed that the corner modes are stable against disorders which breaks the mirror symmetries, there is still obvious difference between the Science paper and the present one. The Science paper used an Anderson type onsite disorder, so the mirror symmetries are preserved "on average". However, the perturbations in the present paper violate the mirror symmetries even on average. Quantitatively, Fig.S4 b shows that the change of the frequency of the corner mode is about 30/2150=1.4%, which has the same order of the perturbation, which is 3%. Therefore, the result cannot be deemed as an evidence of robustness against such kind of perturbation. For Fig.S4a, I think more discussions are also needed to clarify why the frequency of corner modes are unchanged.
The authors realize an acoustic metamaterial with a quantized octupole moment. The paper is well written, the data conclusively shows the expected spectroscopic characteristics of a quantized octupole insulator and thus I recommend publication of this work in Nature Communications.
I have a couple of comments: In establishing the existence of a hierarchy of multipole moments in the present work, a more careful description is needed. For example, in the abstract, the authors say that "We directly observe 0D corner states, 1D hinge states, 2D surface states, and 3D bulk states". Putting the 0D corner states on equal footing as the hinge, surface, and bulk states is a bit misleading for the following reason: in an octupole insulator -and in the presence of chiral (sublattice) symmetry-only the 0D corner states are topologically protected. Hinge and surface modes are gapped in an octupole insulator and in principle can fuse with the bulk states without altering the octupole topology. The manuscript should correct statements in relation to this discussion in various parts of it.
I would recommend the authors to reconsider including the sentence "From this perspective, the previously-realized quadrupole TI phases are 2D projections of 3D octupole TIs". It is not clear what "projection" means in this context.
In the sentence "The unit cell can be regarded as two couple unit cells of quadrupole TIs with opposite settings", the term "opposite settings" is obscure, please clarify.
In the sentence "...meaning that there are only bulk states and no topologically guaranteed boundary states", it should say "corner states", not "boundary states", as only corner states are topologically protected.
The authors should discuss the role of chiral symmetry in the protection of the corner states. Only under chiral symmetry, a nontrivial quantized octupole moment will have mid-gap corner states. This should be particularly emphasized in this system as they seem to achieve nearly perfect chiral symmetry in the setup, which is reflected in the symmetry of the spectrum in Fig. 3c.
In the caption of Fig 1, second sentence, the description of various cases with parenthesis is confusing, as they refer to two sets of different things. Separating this sentence into two sentences will solve the confusion.
Reviewer #3 (Remarks to the Author): The work demonstrates the realization of a model known to be a higher-order topological insulator with corner states and a quantized octupole moment in a classical setup, namely as a system of coupled acoustic resonators arranged into a 3D lattice.
I have no reason to question the quality of the experimental setup or the measurements. The material is clearly presented as well.
However, I think that what is studied experimentally is neither topological nor quantized and these essential shortcomings render the results not relevant for publication in nature communication.
Topology: The topological bulk-boundary correspondence implies that one can predict, based on a bulk topological invariant taking a nontrivial value, that there will be corner states in an open geometry (in the case at hand). However, for the model considered, this bulk-boundary-correspondence requires certain symmetries to hold (both in the bulk and in the open geometry). For instance, if as C3 rotation symmetry around the (1,1,1) axis is preserved (i.e., with respect to the axis that goes through the corner in the open geometry case), this can guarantee the bulk-boundary correspondence. However, this is not discussed in the manuscript at all. Instead, mirror symmetries are mentioned, but these rather guarantee the quantization of nested Wilson loop spectra and not the bulk-boundary correspondence. Some evidence toward the topological stability of the corner modes in the spirit discussed here is given at the end of the SI, but I think the explanation is not enough to highlight the important role played by spatial symmetry protecting the topology. Also, the C3 symmetry I refer to above is evidently broken by the physical structure of the system. Thus, it remains unclear to me what symmetry actually preserves topological bulk-boundary correspondence in the case at hand, if any.
Quantization: The electronic counter-part to this system is called a quantized octupole insulator because the octupole moment of the electronic charge distribution in the unit cell is quantized. This is a measurable observable. In the acoustic case, I do not see any corresponding quantized observable. Certainly, none has been measured in the experiment. Of course, the multiply nested Wilson loop is quantized as explained in the manuscript and SI, but what bulk observable corresponds to it? Short of a filled Fermi see in a classical system, I do not see any. In analogy, classical realizations of Chern bands (of which there are several) are also not called realizations of quantum Hall states, because there is no quantized Hall conductivity. The measurement of some bulk topological quantity would have been desirable. In view of this, I find the terminology "quantized…" misleading for the case at hand.
As a final small remark, I would have found a demonstration of the exponential decay of the corner modes and a comparison of the decay strength with the one expected from the ratio \gamma/\lambda useful for completeness.
I think that these points on topology and quantization are so severe and fundamental to the approach taken by the authors that I do not see how they can be overcome in a revision.
We are grateful for the constructive comments on this manuscript (NCOMMS-19-39046-T) from all the reviewers.
In the text below each of the comments from each reviewer is quoted in italics and followed by the corresponding detailed response. We have also revised the manuscript and the Supplementary Information accordingly, and these updates are highlighted in red in those files.
In the text below, the references to these updates are also highlighted in red.
This is an experimental paper in which the authors designed, fabricated and characterized a 3D acoustic metamaterial which is a quantized octupole TI. They observed 0D corner states, 1D hinge states, 2D surface states, and 3D bulk states, representing the topological hierarchy of quantised octupole, quadrupole and dipole moments. This is a nice piece of work. They gave a rigorous treatment for the tight-binding model and they implemented the model with a simple and clever design that achieved negative hopping. They gave control experiments of trivial phases which makes their case more convincing. The presentation is clear and concise. I think the paper amply meets the expectation of Nature Commun. I only find some rather minor issues which the authors might want to address.

Authors Response:
We thank Reviewer #1 for the high opinions and encouraging comments.

Reviewer Comments:
1. According to my understanding, whether the coupling is positive or negative (as illustrated in Fig.1d) depends on the convention of the phase of the dipolar eigenstate in each cavity. There is no reason why I cannot regard the two modes in the right panel as "positive coupling", but the two coupled modes in the left panel as "negative coupling". This new convention is equivalent to the original one up to a gauge transformation. In the full structure, there are 4 cavities in each unit cell, and the cavities have different orientations. The signs of the coupling bonds are not obvious unless all the phases of the dipolar cavity modes are fixed. Therefore, I think the authors should give some explanation about the their convention more explicitly, and clarify that the topology of the system does not depend on the convention of the gauge.

Authors Response:
We thank Reviewer #1 for pointing out this issue. We agree that whether the coupling is positive or negative depends on the convention. To make our convention more explicitly, we have revised Fig. 1d and added more explanation in the main text as: "Here the sign of a coupling in the lattice can be determined by looking at the configuration of associated resonators and connecting waveguide. There are only two possible in-plane configurations as shown in Fig. 1d where the connecting waveguides are either located at the upper part (configuration A) or lower part (configuration B). Throughout this paper we assume positive (negative) coupling is implemented by configuration A (B). Note that one can also assume positive (negative) coupling is implemented by configuration B (A), which corresponds to a gauge transformation and thus does not alter the topology of the system." Reviewer Comments: 2. Since the fields inside the cavities have inhomogeneous distributions, it is not obvious how the acoustic intensity at each site shown in Fig. 2,3,4 is defined. Do they refer to the field intensity at a certain point in the cavity or to the average intensity inside the whole cavity?

Authors Response:
The acoustic intensity is measured at the side of each resonator where the field is at the maxima. As explained in the Methods part, there are two small holes at two side of each resonator for excitation and detection. When doing experiment, we insert the microphone into one of the holes and place it just at the resonance's maxima to collect the signal. We have added one more sentence in the Methods part as "…… and collected by a microphone (Brüel&Kjaer Type 4182) which is placed at the maximum point of the dipole resonance." to clarify this point. Fig. S4 violate the underlying symmetries (the two anti-commutative mirror symmetries) that protect the quantization of the octupole momentum. Symmetry-protected topological phases are only stable against the perturbations preserving the underlying symmetries. If the symmetry constraints are compromised, a system can continuously change from a SPT nontrivial phase to a SPT trivial phase without gap closing. Though the original Science paper, "quantized electric multipole insulators", also claimed that the corner modes are stable against disorders which breaks the mirror symmetries, there is still obvious difference between the Science paper and the present one. The Science paper used an Anderson type onsite disorder, so the mirror symmetries are preserved "on average". However, the perturbations in the present paper violate the mirror symmetries even on average. Quantitatively, Fig.S4 b shows that the change of the frequency of the corner mode is about 30/2150=1.4%, which has the same order of the perturbation, which is 3%. Therefore, the result cannot be deemed as an evidence of robustness against such kind of perturbation. For Fig.S4a, I think more discussions are also needed to clarify why the frequency of corner modes are unchanged.

Authors Response:
We agree that the two types of perturbations in Fig. S4 break the mirror symmetries that protect the quantization of octupole moment. We also agree that the term 'robustness' is not very suitable for describing the situations here. In fact, the main purpose of section D in the Supplementary Information is to study the stability of the corner modes under local perturbations, which is meaningful for potential applications. We have revised Supplementary Information D accordingly to remove the term "robustness" and emphasized the motivation of this section by saying "As can be seen in Figs. S4a-c, the corner states are sublattice polarized. This gives a unique stability to the corner states under some local perturbations." Although both perturbations break the mirror symmetries, perturbation 1 acts only on the three sites next to the corner. Since the corner state is sublattice-polarized, this perturbation would have no effects on the corner state. We have added one more sentence "As can be seen, the corner state is almost unaffected since the corner state has neglectable distribution on the perturbed sites." in Supplementary Information D to clarify this point. In contrast, perturbation 2 acts on the corner site and thus shifts the frequency of corner state. This different responses to different kinds of perturbations could be useful in applications like sensing. To address this point, we have added one sentence in Supplementary Information D as "This unique stability of corner states under local perturbations may be useful for further applications in sensing devices." The authors realize an acoustic metamaterial with a quantized octupole moment. The paper is well written, the data conclusively shows the expected spectroscopic characteristics of a quantized octupole insulator and thus I recommend publication of this work in Nature Communications.

Authors Response:
We thank Reviewer #2 for the recommendation.

Reviewer Comments: I have a couple of comments:
In establishing the existence of a hierarchy of multipole moments in the present work, a more careful description is needed. For example, in the abstract, the authors say that "We directly observe 0D corner states, 1D hinge states, 2D surface states, and 3D bulk states". Putting the 0D corner states on equal footing as the hinge, surface, and bulk states is a bit misleading for the following reason: in an octupole insulator -and in the presence of chiral (sublattice) symmetryonly the 0D corner states are topologically protected. Hinge and surface modes are gapped in an octupole insulator and in principle can fuse with the bulk states without altering the octupole topology. The manuscript should correct statements in relation to this discussion in various parts of it.

Authors Response:
We thank Reviewer #2 for the insightful comments and agree that putting corner states together with hinge, surface and bulk states may mislead readers to feel that the hinge and surface states are also topologically protected. To remove the potential confusion, we have revised the abstract as well as other parts where corner, hinge, surface and bulk states are put on the same footing. The term "topological" is put in front of corner states and the term "gapped" is put in front of hinge, surface and bulk states. Besides, we have also added the sentence "Note only the corner states are topologically protected while the hinge and surface states are not." in the main text.

Reviewer Comments:
I would recommend the authors to reconsider including the sentence "From this perspective, the previously-realized quadrupole TI phases are 2D projections of 3D octupole TIs". It is not clear what "projection" means in this context.

Authors Response:
We thank Reviewer #2 for the suggestion and have removed this sentence in the revised manuscript.

Reviewer Comments:
In the sentence "The unit cell can be regarded as two couple unit cells of quadrupole TIs with opposite settings", the term "opposite settings" is obscure, please clarify.

Authors Response:
We have changed the term "opposite settings" to "opposite signs of couplings".

Reviewer Comments:
In the sentence "...meaning that there are only bulk states and no topologically guaranteed boundary states", it should say "corner states", not "boundary states", as only corner states are topologically protected.

Authors Response:
We have revised this sentence accordingly.

Reviewer Comments:
The authors should discuss the role of chiral symmetry in the protection of the corner states. Only under chiral symmetry, a nontrivial quantized octupole moment will have mid-gap corner states. This should be particularly emphasized in this system as they seem to achieve nearly perfect chiral symmetry in the setup, which is reflected in the symmetry of the spectrum in Fig.  3c.

Authors Response:
We agree here chiral symmetry plays an important role in pining the corner states at mid-gap. In the real structure chiral symmetry is approximately presented, as can be seen from the numerical results in Fig. S1 where the next-nearest couplings are shown to be neglectable and the band structure is symmetric. Also this can be seen from experimentally measured spectra, as mentioned by reviewer. We have added the sentence "…which are protected by chiral symmetry that approximately holds in the real structure due to the careful design process." in the revised manuscript to emphasize the role of chiral symmetry.

Reviewer Comments:
In the caption of Fig 1, second sentence, the description of various cases with parenthesis is confusing, as they refer to two sets of different things. Separating this sentence into two sentences will solve the confusion.

Authors Response:
We thank the reviewer for the suggestion. We have rewritten the sentence as: The left (right) panel illustrates the intra-cell (inter-cell) couplings. Here solid and dashed lines representing positive and negative couplings, respectively. The work demonstrates the realization of a model known to be a higher-order topological insulator with corner states and a quantized octupole moment in a classical setup, namely as a system of coupled acoustic resonators arranged into a 3D lattice.
I have no reason to question the quality of the experimental setup or the measurements. The material is clearly presented as well.
However, I think that what is studied experimentally is neither topological nor quantized and these essential shortcomings render the results not relevant for publication in nature communication.

Authors Response:
We thank Reviewer #3 for commenting that "I have no reason to question the quality of the experimental setup or the measurements. The material is clearly presented as well." Below we provide more clarifications on the topology and quantization of our system, which we hope can lift the concerns from the reviewer.

Reviewer Comments:
Topology: The topological bulk-boundary correspondence implies that one can predict, based on a bulk topological invariant taking a nontrivial value, that there will be corner states in an open geometry (in the case at hand). However, for the model considered, this bulk-boundarycorrespondence requires certain symmetries to hold (both in the bulk and in the open geometry). For instance, if as C3 rotation symmetry around the (1,1,1) axis is preserved (i.e., with respect to the axis that goes through the corner in the open geometry case), this can guarantee the bulkboundary correspondence. However, this is not discussed in the manuscript at all. Instead, mirror symmetries are mentioned, but these rather guarantee the quantization of nested Wilson loop spectra and not the bulk-boundary correspondence. Some evidence toward the topological stability of the corner modes in the spirit discussed here is given at the end of the SI, but I think the explanation is not enough to highlight the important role played by spatial symmetry protecting the topology. Also, the C3 symmetry I refer to above is evidently broken by the physical structure of the system. Thus, it remains unclear to me what symmetry actually preserves topological bulk-boundary correspondence in the case at hand, if any.