Observation of a phononic quadrupole topological insulator
Open accessAbstract
The modern theory of charge polarization in solids is based on a generalization of Berry’s phase. The possibility of the quantization of this phase arising from parallel transport in momentum space is essential to our understanding of systems with topological band structures. 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 photonic or phononic crystals. The theory of this quantized polarization has recently been extended from the dipole moment to higher multipole moments. 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 states. 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 guides in higher dimensions, and thereby open up a new path for the design of metamaterials.
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TL;DR
The paper reports the first experimental observation of a phononic quadrupole topological insulator in a mechanical metamaterial, with gapped edge modes and in-gap corner states. It also compares the topologically nontrivial phase to trivial and \((1/2,0)\)-phase samples predicted by quadrupole theory.
Key findings
- In the nontrivial sample with \(\gamma<\lambda\), the measured spectra show edge-dominated response near 72.92 kHz and 74.89 kHz and sharp corner resonances inside the gap; the trivial sample with \(\gamma>\lambda\) shows no edge/corner-dominated regions and no resonances between 72.92 kHz and 74.89 kHz.
- Each of the four corners hosts only one in-gap mode, despite each corner terminating two gapped edges; the four corner resonances are split by next-to-nearest-neighbor couplings that break particle-hole symmetry.
- A sample in the \((p^{\nu_x},p^{\nu_y})=(1/2,0)\) phase shows no in-gap states, with edge weight coming from only two surfaces.
- The extracted reduced model gives \(|\gamma/\lambda|=0.28\), offset frequency \(\nu_0=73.895\pm0.03\) kHz, \(\lambda=6.69\pm0.17\times10^9\ (\mathrm{rad/s})^2\), and \(\gamma=1.89\pm0.07\times10^9\ (\mathrm{rad/s})^2\); the decay length is \(\xi/a\approx1.6\), with about 92% of edge/corner mode energy on the outermost row/corner site.
Methods
- The metamaterial uses 5×5×0.364 mm single-crystal silicon plates vibrating in the first non-rigid \(d_{xy}\) mode with two nodal lines; thin beams mediate positive or negative couplings, and the design was obtained by combinatorial search followed by gradient optimization.
- Measurements were performed on a 10×10 sample: an ultrasound air transducer excited a single plate, and a laser interferometer measured the out-of-plane vibration amplitude; mechanical energy was taken as \(\varepsilon_i\propto\Delta z_i^2\), with filters separating bulk, edge, and corner responses.
- The paper uses nested Wilson loops and Wannier bands to define the quadrupole invariant, plus finite-element simulations and a reduced-order model to extract the effective dynamical matrix and compare with experiment.
Limitations
- The paper states that spurious long-range couplings from off-resonant admixing of other single-plate modes break mirror symmetries, so the polarizations are not precisely quantized and the corner resonances are split.
- The paper states that bulk, edge, and corner modes spectrally overlap, so there is no unique way to separate them in the measurements; the separation is justified by the short decay length \(\xi/a\approx1.6\).
- The paper reports measurement uncertainty of about 11.2 pm on displacements and fabrication errors including thickness variation ≤1 µm, sidewall angle variation ≤2.5°, and crystalline-axis misalignment; evident from the text, the demonstration is limited to a specific 10×10 silicon phononic metamaterial design.
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Claims the paper makes
0 of 19 cited sources checked · 0 supported · 0 flagged
- Findingown data§ Observation of a phononic quadrupole topological insulator · p. 3
The mechanical metamaterial exhibits predicted gapped edge states.
“close to 72.92 kHz and 74.89 kHz, the response is dominated by the edges, indicative of the bulk-induced gapped edge modes.”
Quote in the paperSubstantial claim - Findingown data§ Observation of a phononic quadrupole topological insulator · p. 3
The mechanical metamaterial exhibits predicted in-gap corner states.
“Sharp resonances at the corners appear in the gap region.”
Quote in the paperSubstantial claim - Findingown data§ Observation of a phononic quadrupole topological insulator · p. 1
The paper experimentally characterizes the bulk, edge, and corner physics of a mechanical metamaterial.
“We experimentally characterize the bulk, edge, and corner physics of a mechanical metamaterial”
Quote in the paperSubstantial claim - Findingown data§ Observation of a phononic quadrupole topological insulator · p. 1
The paper provides the first measurements of a phononic quadrupole insulator.
“Here, we provide the first measurements of a phononic quadrupole insulator.”
Quote in the paperSubstantial claim- Rests on an assumption: No experimental observation of a quadrupole insulator existed prior to this work.
- Findingown data§ Observation of a phononic quadrupole topological insulator · p. 3
The bulk induces gapped edge-modes on all four sides of the sample.
“Note that the bulk induces gapped edge-modes on all four sides of the sample.”
Quote in the paperSubstantial claim
- Findingnone given§ Methods · p. 11
The plate and beam geometry of Fig. 1d implements the sought after weak and strong, positive and negative coupling matrix elements.
“The plate and beam geometry of Fig. 1d implements the sought after weak and strong, positive and negative coupling matrix elements.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: Each degree of freedom in the objective model is mapped to a single plate, which vibrates in its first non-rigid-body mode with dxy symmetry.
- Findingown data§ Observation of a phononic quadrupole topological insulator · p. 3
In the trivial phase, no frequency range is dominated by the edge or corner response.
“No frequency range is dominated by the edge or corner response.”
Quote in the paperSubstantial claim - Findingown data§ Methods · p. 7
Along the edges, the measured decay of the wave function follows the theoretical prediction with a decay length ξ/a≈1.6.
“Along the edges we show the decay of the wave function and compare their envelope to the theoretical prediction with a decay length ξ/a ≈ 1 . 6.”
Quote in the paperNot a substantial claim - Findingnone given§ Methods · p. 9
We combine geometric elements (silicon plates, beams and holes) to create a material that reproduces the discrete model of Benalcazar et al. over a range of frequencies.
“We combine geometric elements (silicon plates, beams and holes) to create a material that reproduces the discrete model of Benalcazar et al. over a range of frequencies.”
Quote in the paperSubstantial claim- The authors state this as an assumption: Each degree of freedom in the objective model is mapped to a single plate, which vibrates in its first non-rigid-body mode with dxy symmetry.
- Findingown data§ Observation of a phononic quadrupole topological insulator · p. 3
Each corner hosts only one resonance peak.
“However, each corner hosts only one resonance peak.”
Quote in the paperSubstantial claim - Findingown data§ Observation of a phononic quadrupole topological insulator · p. 3
In the trivial phase, the relative weight of the three curves is in good accordance with the respective number of sites in the bulk, edges, and corners.
“the relative weight of the three curves is in good accordance with the respective number of sites in the bulk, edges, and corners, respectively.”
Quote in the paperSubstantial claim - Findingown data§ Methods · p. 9
The resulting sign change is different (inside vs. between unit-cells) on the two edges.
“The resulting sign change is indeed different (inside vs. between unit-cells) on the two edges.”
Quote in the paperSubstantial claim - Findingreasoning§ Observation of a phononic quadrupole topological insulator · p. 4
The phenomenology of in-gap corner modes is still observed.
“the phenomenology of in-gap corner modes is still observed”
Quote in the paperSubstantial claim - Findingown data§ Methods · p. 6
As the bulk, edge, and corner modes spectrally overlap, there is no unique way to separate them in the measurements.
“As the bulk, edge, and corner modes spectrally overlap, there is no unique way to separate them in our measurements.”
Quote in the paperNot a substantial claim - Findingreasoning§ Observation of a phononic quadrupole topological insulator · p. 4
The polarizations are not precisely quantized.
“the polarizations are not precisely quantized.”
Quote in the paperNot a substantial claim - Findingcitation§ Observation of a phononic quadrupole topological insulator · p. 4
The paper leveraged perturbative metamaterials to find a first implementation of a quantized quadrupole insulator.
“we leveraged this technique to find a first implementation of a quantized quadrupole insulator, a new class of topological materials.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [28] Matlack, K. H ., Serra-Garcia, M ., Palermo, A ., Huber, S. D . & Daraio, C. Designing perturbative metamaterials from discrete models. Nat. Mater. https://doi.org/10.1038/s41563-017-0003-3 (2018) source
Not checked in the free run.
- Rests on an assumption: No experimental observation of a quadrupole insulator existed prior to this work.
- Findingown data§ Methods · p. 9
The decay of the edge modes follows the expected decay with ξ/a≈1.6.
“Finally, to further justify our filtering, we show that also the decay of the edge modes follows the expected decay with ξ/a ≈ 1 . 6.”
Quote in the paperNot a substantial claim - Findingown data§ Observation of a phononic quadrupole topological insulator · p. 3
In the trivial sample, no resonances appear in the gap between 72.92 kHz and 74.89 kHz.
“No resonances appear in the gap between 72.92 kHz and 74.89 kHz.”
Quote in the paperSubstantial claim - Findingreasoning§ Observation of a phononic quadrupole topological insulator · p. 4
The absence of an exact My symmetry leads to a breaking of the rule ν+x(ky)+ν-x(ky)=1/2.
“The absence of an exact M y symmetry indeed leads to a breaking of this rule.”
Quote in the paperNot a substantial claim - Findingown data§ Observation of a phononic quadrupole topological insulator · p. 4
Spurious long-range couplings mediated by off-resonant admixing of other single-plate modes induce a certain amount of mirror-symmetry breaking.
“spurious long-range couplings mediated by off-resonant admixing of other single-plate modes induce a certain amount of mirror-symmetry breaking.”
Quote in the paperNot a substantial claim - Findingown data§ Observation of a phononic quadrupole topological insulator · p. 4
The nearest neighbor couplings in the reduced model follow the blueprint of the target model.
“The nearest neighbor couplings indeed follow the blueprint of the target model shown in Fig. 1c.”
Quote in the paperSubstantial claim - Findingreasoning§ Methods · p. 7
Transitions away from the C4-symmetric line happen through bulk-induced edge-transitions, where no bulk gap is closing.
“The transitions away from the C 4 -symmetric line happen through bulk-induced edge-transitions, where no bulk gap is closing.”
Quote in the paperSubstantial claim - Findingown data§ Methods · p. 7
The second transition is induced by a gap closing at 0.
“The second transition is induced by a gap closing at 0.”
Quote in the paperNot a substantial claim - Findingreasoning§ Methods · p. 8
Given the decay length ξ≈1.6a, the residual weight of maximally 2% on the corners other than the ones excited is stemming from spurious acoustic excitation rather than hybridization.
“Given the decay length ξ ≈ 1 . 6 a , where a is the lattice constant, the residual weight of maximally 2% on the corners other than the ones excited is stemming from spurious acoustic excitation rather than hybridization.”
Quote in the paperSubstantial claim - Findingown data§ Methods · p. 7
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 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.”
Quote in the paperNot a substantial claim - Study design and conductown data§ Observation of a phononic quadrupole topological insulator · p. 1
The paper corroborates its findings by comparing the mechanical properties of a topologically non-trivial system to samples in other phases predicted by the quadrupole theory.
“We further corroborate our findings by comparing the mechanical properties of a topologically non-trivial system to samples in other phases predicted by the quadrupole theory.”
Quote in the paperNot a substantial claim - Premiseown data§ Observation of a phononic quadrupole topological insulator · p. 2
The dimerized hopping model with amplitudes λ and γ leads to a bandgap between two pairs of degenerate bands for λ=γ.
“The dimerized hopping with amplitude λ and γ leads to a bandgap between two pairs of degenerate bands for λ = γ”
Quote in the paperSubstantial claim - Study design and conductown data§ Methods · p. 6
The separation using the filters shown in Fig. 2d, which select sites in the interior, along the edge and the corner sites respectively, is well justified because the decay length is extremely short (ξ/a≈1.6).
“However, the fact that the decay length is extremely short ( ξ/a ≈ 1 . 6, see above), a separation using the filters shown in Fig. 2d, where we simply select sites in the interior, along the edge and the corner sites respectively, is well justified.”
Quote in the paperSubstantial claim - Premisecitation§ Methods · p. 6
The presence of two mirror symmetries Mx and My that do not commute is a necessary requirement for the nested polarizations pν±x and pν±y to be quantized to 0 or 1/2.
“It can be shown that the presences of two mirror-symmetries M x and M y that do not commute are a necessary requirement for the nested polarizations p ν ± x and p ν ± y to be quantized to 0 or 1 / 2. 14”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [14] Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017) source
Not checked in the free run.
- The authors state this as an assumption: Two bands n=1,2 are filled.
- Study design and conductreasoning§ Methods · p. 6
The termination is such that all plates see an identical surrounding, independent of their location in bulk, along the edges or on the corners.
“Moreover, our termination is such that all plates see an identical surrounding, independent of their location in bulk, along the edges or on the corners.”
Quote in the paperNot a substantial claim - Study design and conductreasoning§ Methods · p. 6
For both viscoelasticity and air dissipation, all disconnected plates suffer from the same damping.
“For both cases all disconnected plates suffer from the same damping.”
Quote in the paperSubstantial claim - Study design and conductown data§ Observation of a phononic quadrupole topological insulator · p. 3
Measurements of the edge-Greens function further support the claim that each corner hosts only one in-gap mode.
“measurements of the edge-Greens function further support this claim”
Quote in the paperSubstantial claim - Premisecitation§ Observation of a phononic quadrupole topological insulator · p. 3
When C4 symmetry is broken by allowing different hoppings in x- and y-direction, the (1/2,0) phase can be reached via a gap-closing of the surface modes.
“When the C 4 -symmetry is broken by allowing for different hoppings in x - and y -direction (see Methods), the phase ( p ν x , p ν y ) = (1 / 2 , 0) can be reached via a gap-closing of the surface modes.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [14] Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017) source
Not checked in the free run.
- Interpretationown data§ Observation of a phononic quadrupole topological insulator · p. 3
The resonances in the four corners are split by the presence of next-to-nearest neighbor couplings that break the particle-hole symmetry.
“The resonances in the four corners are split by the presence of next-to-nearest neighbor couplings that break the particle-hole symmetry.”
Quote in the paperNot a substantial claim - Study design and conductreasoning§ Methods · p. 6
The perturbative nature of the beams restricts the effects of the couplings on the dissipation.
“The perturbative nature of our beams (recall the bandwidth of ∼ 5 kHz around the center frequency of ∼ 74 kHz), restricts also the effects of the couplings on the dissipation.”
Quote in the paperSubstantial claim - Interpretationcitation§ Methods · p. 6
The decay of edge and corner states into the bulk is exponentially decaying with a decay length ξ/a = 2/(log|λ/γ|) (in units of the site-to-site distance a), in analogy to the Su-Schrieffer-Heeger model.
“The decay of the edge and corner states into the bulk is simple to derive in analogy to the Su-Schrieffer-Heeger model, 30 where the wavefunction has a node on every other site and is exponentially decaying with a decay length of ξ/a = 2 / (log | λ/γ | ) (in units of the site-to-site distance a ).”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [30] Imhof, S. et al. Topoelectrical circuit realization of topological corner modes. Preprint at https://arxiv.org/abs/1708.03647 (2017)
Not checked in the free run.
- Premisecitation§ Observation of a phononic quadrupole topological insulator · p. 2
If Wannier bands are gapped, the eigenvectors of the Wilson loops define the bulk-induced edge polarization of the bands below the gap.
“If these Wannier bands ν ± ( k α ) are gapped, the eigenvectors of the Wilson loops can be used to define the bulk-induced edge polarization of the bands below the gap p ν -α”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [14] Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017) source
Not checked in the free run.
- Premisecitation§ Observation of a phononic quadrupole topological insulator · p. 2
The presence of inversion symmetry I and non-commuting mirror symmetries Mx and My leads to a well-defined and quantized edge polarization pν-α ∈ {0, 1/2}.
“the presence of inversion symmetry I and non-commuting mirror symmetries M x and M y lead to a well defined and quantized p ν -α ∈ { 0 , 1 / 2 }”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [5] Kane, C. L. & Mele, E. J. Z2 topological order and the quantum spin hall effect. Phys. Rev. Lett. 95, 146802 (2005) source
Not checked in the free run.
- Premisecitation§ Observation of a phononic quadrupole topological insulator · p. 2
In the quadrupole insulator, each corner hosts only one mode.
“it is an important hall-mark of the bulk nature of the quadrupole insulator that each corner hosts only one mode”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [14] Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017) source
Not checked in the free run.
- Premisereasoning§ Observation of a phononic quadrupole topological insulator · p. 2
The π-flux requires mirror symmetry My to be accompanied by a gauge transformation, leading to non-commutation of Mx and My.
“This π -flux requires the mirror-symmetry around the horizontal axis ( M y ) to be accompanied by a gauge-transformation, leading to the non-commutation of M x and M y .”
Quote in the paperNot a substantial claim - Premisecitation§ Methods · p. 6
The eigenvalues ν±(ky) of the Wilson-loop operator Wx(ky) are in one-to-one correspondence to the spectrum of an edge perpendicular to the x-coordinate (or perpendicular to y when x and y are interchanged).
“The eigenvalues ν ± ( k y ) of W x ( k y ) are in one-to-one correspondence to the spectrum of an edge perpendicular to the x -coordinate 27 (or perpendicular to y when x and y are interchanged).”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [27] Fidkowski, L., Jackson, T. S. & Klich, I. Model characterization of gapless edge modes of topological insulators using intermediate Brillouin-zone functions. Phys. Rev. Lett. 107, 036601 (2011) source
Not checked in the free run.
- The authors state this as an assumption: Two bands n=1,2 are filled.
- Interpretationnone given§ Observation of a phononic quadrupole topological insulator · p. 3
A small mirror symmetry breaking leads to the non-degeneracy of the in-gap states.
“A small mirror symmetry breaking leads to the non-degeneracy of the in-gap states”
Quote in the paperNot a substantial claim - Premisereasoning§ Methods · p. 10
The modes of isolated plates hybridize to Bloch bands of small bandwidth, preventing bands originating from unwanted modes to cross in frequency.
“First, the modes of isolated plates hybridize to Bloch bands of small bandwidth, preventing bands originating from unwanted modes to cross in frequency.”
Quote in the paperSubstantial claim - Study design and conductown data§ Methods · p. 8
The recorded response (amplitude and phase) allows to reconstruct the eigenfunctions ψ(x,y) of the individual corner modes.
“The recorded response (amplitude and phase) allows to reconstruct the eigenfunctions ψ ( x, y ) of the individual corner modes.”
Quote in the paperSubstantial claim
- Classified as probably not a substantial claim. 23
- May combine two claims. 19
Probably not a substantial claim: may not pass S3 (load-bearing).
Findingreasoning§ Methods · p. 6For the ratio λ/γ≈0.28, the decay length is ξ/a≈1.6.
“For our ratio of of λ/γ ≈ 0 . 28 (see below) this results in ξ/a ≈ 1 . 6.”
Quote in the paperNot a substantial claimMay combine two claims.
Findingreasoning§ Methods · p. 11The resulting matrix K has the same eigenfrequencies and projected eigenmodes as the full system and therefore provides a good description of the system's dynamics.
“The resulting matrix K has the same eigenfrequencies and projected eigenmodes as the full system and therefore provides a good description of the system's dynamics.”
Quote in the paperSubstantial claim- The authors state this as an assumption: Each degree of freedom in the objective model is mapped to a single plate, which vibrates in its first non-rigid-body mode with dxy symmetry.
May combine two claims.
Findingown data§ Methods · p. 7The four corner modes are independent and the spread in their frequency arises not from their hybridization.
“The four panels proof that the four corner modes are independent and the spread in their frequency arises not from their hybridization.”
Quote in the paperSubstantial claimMay combine two claims.
Findingown data§ Methods · p. 9The edge with positive couplings has nodes between the unit cells, whereas the edge with negative couplings has nodes inside the unit cell, establishing a direct measurement of the negative couplings giving rise to the π-flux.
“The edge with positive couplings has nodes between the unit cells, whereas the edge with negative couplings has nodes inside the unit cell, establishing a direct measurement of the negative couplings giving rise to the π -flux.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ Observation of a phononic quadrupole topological insulator · p. 4The symmetry breaking terms do not lead to any gap-closing, neither on the edge nor in the bulk.
“the symmetry breaking terms do not lead to any gapclosing, neither on the edge nor in the bulk.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ Methods · p. 11This results in an offset frequency for our mode of ν0=73.895±0.03 kHz and coupling matrix elements λ=6.69±0.17×10^9 (rad/s)^2 and γ=1.89±0.07×10^9 (rad/s)^2.
“This results in an offset frequency for our mode of ν 0 = 73 . 895 ± 0 . 03 kHz and the coupling matrix elements are given by λ = 6 . 69 ± 0 . 17 × 10 9 (rad / s) 2 and γ = 1 . 89 ± 0 . 07 × 10 9 (rad / s) 2”
Quote in the paperSubstantial claim- The authors state this as an assumption: Each degree of freedom in the objective model is mapped to a single plate, which vibrates in its first non-rigid-body mode with dxy symmetry.
May combine two claims.
Findingnone given§ Methods · p. 6The eigenvalues of D(kx,ky) are given by ζ=±|d(k)|, leading to two doubly-degenerate bands.
“The eigenvalues of D ( k x , k y ) are given by ζ = ±| d ( k ) | , leading to two doubly-degenerate bands.”
Quote in the paperSubstantial claimMay combine two claims.
Findingown data§ Methods · p. 7The brown area marks the quantized quadrupole phase, 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 brown area marks the quantized quadrupole phase, 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.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ Methods · p. 6Bulk gap-closings occur when d(k)=0, which only happens for the C4-symmetric case at λ=±γ.
“Bulk gap-closings occur when d ( k ) = 0, which only happens for the C 4 -symmetric case at λ = ± γ .”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Findingreasoning§ Methods · p. 61-e^{-4/ξ}≈92% of the energy of an edge/corner mode is stored on the outermost row/corner site.
“In other words, 1 -e -4 /ξ ≈ 92% of the energy of a edge/corner mode is stored on the outermost row/corner site.”
Quote in the paperNot a substantial claimMay combine two claims.
Findingown data§ Observation of a phononic quadrupole topological insulator · p. 3Measurements on a sample in the (1/2,0) phase show no in-gap states and the edge-dominated frequency region draws its weight from only two surfaces.
“In Fig. 3e, we show measurements on a sample in the (1 / 2 , 0)-phase, where no in-gap states appear and the frequency region dominated by the edges draws its weight from only two surfaces.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Findingreasoning§ Methods · p. 7The dashed line indicates the C4-symmetric line, where the bulk gap is closing at the phase transition.
“The dashed line indicates the C 4 -symmetric line, where the bulk gap is closing at the phase transition.”
Quote in the paperNot a substantial claimMay combine two claims.
Premisecitation§ Observation of a phononic quadrupole topological insulator · p. 1The surface dipole moments indicate gapped edge modes while the corner charges motivate in-gap corner excitations.
“The former is indicating gapped edge modes while the latter motivates the presence of in-gap corner excitations”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [14] Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017) source
Not checked in the free run.
May combine two claims.
Premisecitation§ Observation of a phononic quadrupole topological insulator · p. 1A bulk quadrupole moment in a finite two-dimensional sample gives rise to surface dipole moments on its one-dimensional edges and uncompensated charges on its zero-dimensional corners.
“A bulk quadrupole moment in a finite two-dimensional sample gives rise to surface dipole moments on its one-dimensional edges as well as to uncompensated charges on the zerodimensional corners”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [14] Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017) source
Not checked in the free run.
Probably not a substantial claim: may not pass S4 (based).
Premisenone given§ Methods · p. 11Surfaces have to be compatible with the unit-cells, i.e., are not allowed to cut through unit-cells.
“Surfaces have to be compatible with the unit-cells, i.e., are not allowed to cut through unit-cells.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisecitation§ Methods · p. 10The weak interaction allows us to approximate the effect of different geometric elements by adding up individual contributions, resulting in a drastic speedup of the calculation times.
“Second, the weak interaction allows us to approximate the effect of different geometric elements by adding up individual contributions (See Ref. 28 for details), resulting in a drastic speedup of the calculation times.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [28] Matlack, K. H ., Serra-Garcia, M ., Palermo, A ., Huber, S. D . & Daraio, C. Designing perturbative metamaterials from discrete models. Nat. Mater. https://doi.org/10.1038/s41563-017-0003-3 (2018) source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing).
Study design and conductown data§ Methods · p. 11The overall total thickness variation within each wafer used is ≤1 μm.
“we confirm that the overall total thickness variation within each wafer we use is ≤ 1 µ m.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Study design and conductreasoning§ Methods · p. 6All measurements are subject to a systematic uncertainty of the interferometer of ∼5 pm and a statistical error of ∼10 pm, resulting in an error estimation on the displacements of ∼11.2 pm.
“All measurements are subject to a systematic uncertainty of the interferometer of ∼ 5 pm, and a statistical error determined by repeated measurements of ∼ 10 pm, resulting in an error estimation on the displacements of ∼ 11 . 2 pm.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Study design and conductreasoning§ Methods · p. 11Variation in feature sizes, when comparing front side to backside, may be up to 32 μm, corresponding to an error of 0.3% for the width of the plates.
“Hence, variation in feature sizes, when comparing front side to backside, may be up to 32 µ m. For the width of the plates this corresponds to an error of 0.3%.”
Quote in the paperNot a substantial claimMay combine two claims.
Premisecitation§ Observation of a phononic quadrupole topological insulator · p. 3The (1/2,0) phase is characterized by gapped edge spectra on two parallel edges and no emergent edge physics on the perpendicular surfaces.
“The (1 / 2 , 0)-phase is characterized by gapped edge spectra on two parallel edges and no emergent edge physics on the perpendicular surfaces.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [14] Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017) source
Not checked in the free run.
Probably not a substantial claim: may not pass S4 (based).
Premisenone given§ Methods · p. 12The impedance miss-match between the steel plates and the wafer leads essentially to fixed boundary conditions Δz=0.
“The impedance miss-match between the steel plates and the wafer leads essentially to fixed boundary conditions ∆ z = 0.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based).
Premisenone given§ Methods · p. 11The coupling matrix elements are given by the ratio of the effective mass-density ρeff of the mode we use and the beam stiffness connecting two plates.
“The coupling matrix elements are given by the ratio of the effective mass-density ρ eff of the mode we use and the beam stiffness connecting two plates.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Study design and conductreasoning§ Methods · p. 11Misalignment of the array with the material crystalline axis (100) results in an overall error of less than 0.1% in the Youngs modulus.
“In either case, this results in an overall error of less than 0.1% in the Youngs modulus.”
Quote in the paperNot a substantial claimMay combine two claims.
Premisecitation§ Observation of a phononic quadrupole topological insulator · p. 3In the (1/2,0) phase, the induced edge modes are in a trivial state and no corner charges are induced.
“Moreover, the induced edge modes are in a trivial state and no corner charges are induced.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [14] Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017) source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing).
Study design and conductnone given§ Methods · p. 6Spectra based on data which is not normalized are almost identical to the ones shown in this paper.
“Moreover, spectra based on data which is not normalized are almost identical to the ones shown in this paper (not shown).”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based).
Premisenone given§ Methods · p. 10For sufficiently good spectral separation and sufficiently weak interactions, a single-mode local basis is enough to capture the material response with high precision.
“For sufficiently good spectral separation and sufficiently weak interactions, a single-mode local basis is enough to capture the material response with high precision.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: Each degree of freedom in the objective model is mapped to a single plate, which vibrates in its first non-rigid-body mode with dxy symmetry.
Probably not a substantial claim: may not pass S3 (load-bearing).
Study design and conductown data§ Methods · p. 11Different sidewall angles between different parts of the wafer are measured to be smaller than 2.5°.
“Different sidewall angles between different parts of the wafer - we measure it to be smaller than 2 . 5 ◦ .”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing), S4 (based).
Interpretationnone given§ Observation of a phononic quadrupole topological insulator · p. 4The platform of a continuous elastic medium provides a direct route to technological applications for any theoretical idea which can be represented by a tight-binding model.
“the platform of a continuous elastic medium provides a direct route to technological applications for any theoretical idea which can be represented by a tight-binding model.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Study design and conductown data§ Methods · p. 6The 0.46 dB variations are negligible with respect to the variations in response of 80 dB.
“The 0.46 dB variations are negligible with respect to the variations in response of 80 dB.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based).
Premisenone given§ Methods · p. 11The definition of γ as the hopping strength inside a unit cell and λ between unit cells renders γ<λ the non-trivial phase.
“The definition of γ as the hopping strength inside a unit cell and λ between unit cells renders γ < λ the non-trivial phase.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ Observation of a phononic quadrupole topological insulator · p. 1When appropriately stacked into three dimensions, the corner modes give rise to chiral one-dimensional modes along edges of the three-dimensional sample.
“When appropriately stacked into three dimensions, the corner modes give rise to chiral one dimensional modes along edges of the three dimensional sample.”
Quote in the paperNot a substantial claim0 of 5 sources checked- cites [17] Huber, S. D. Topological mechanics. Nat. Phys. 12, 621–623 (2016) source
Not checked in the free run.
- cites [23] Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013) source
Not checked in the free run.
- cites [24] Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Taylor, J. M. Imaging topological edge states in silicon photonics. Nat. Photon. 7, 1001–1005 (2013) source
Not checked in the free run.
- cites [25] Lang, N. & Büchler, H. P. Topological networks for quantum communication between distant qubits. Quantum Inf. 3, 47 (2017) source
Not checked in the free run.
- cites [26] Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators. Phys. Rev. B 96, 245115 (2017) source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Study design and conductown data§ Methods · p. 11Total thickness variation is smaller than 1 μm, corresponding to less than a 0.3% variation across the wafer.
“Total thickness variation - which we characterize being smaller than 1 µ m, and therefore stands for less than a 0.3% variation across the wafer.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Study design and conductown data§ Methods · p. 6Careful error-propagation analysis results in error bars on all the figures which are smaller than the symbol size.
“Careful errorpropagation analysis results in error bars on all the figures which are smaller than the symbol size.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Study design and conductreasoning§ Methods · p. 11We can use the same design of Fig. 1d and realize all phases shown in this paper by starting from a 10×10 sample in the (1/2,1/2) phase, then move the cut in y-direction by one row of sites to reach the (1/2,0)-phase, and finally move the termination one column to end up in the (0,0)-phase.
“In turn, this also means we can use the same design of Fig. 1d and realize all phases shown in this paper by starting from a 10 × 10 sample in the (1 / 2 , 1 / 2)phase, then move the cut in y -direction by one row of sites to reach the (1 / 2 , 0)-phase. Finally we move the termination one column and end up in the (0 , 0)-phase.”
Quote in the paperNot a substantial claimMay combine two claims.
Premisereasoning§ Observation of a phononic quadrupole topological insulator · p. 2The model has inversion I and C4 rotational symmetry forcing pν-x = pν-y, and particle-hole symmetry fixing corner modes to the middle of the gap.
“The present model also has inversion I and C 4 rotational symmetry (again up to a gauge-transformation) forcing p ν -x = p ν -y as well as particle-hole symmetry fixing the corner modes to the middle of the gap.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Premisecitation§ Observation of a phononic quadrupole topological insulator · p. 2Nested Wilson loops yield a quantized quadrupole moment.
“Benalcazar et al. 14 proposed to use nested Wilson loops as a way to obtain a quantized quadrupole moment”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [14] Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017) source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing).
Study design and conductown data§ Methods · p. 6The transducer has an essentially flat frequency response over the frequencies of interest.
“The transducer has an essentially flat frequency response over the frequencies of interest, see Extended Data figure 2 (measured with a second air-transducer).”
Quote in the paperNot a substantial claim
The paper relies on this without support.
No experimental observation of a quadrupole insulator existed prior to this work.
“However, such a state of matter has not been observed experimentally.”
The authors state this as an assumption.
All modes suffer from the same loss, or equivalently, have the same quality factor Q≈1000.
“This is only valid under the assumption that all modes suffer from the same loss, or equivalently, have the same quality factor Q ≈ 1000 (determined from the width of the corner modes).”
The authors state this as an assumption.
Each degree of freedom in the objective model is mapped to a single plate, which vibrates in its first non-rigid-body mode with dxy symmetry.
“Each degree of freedom in the objective model is mapped to a single plate, which is assumed to vibrate in its first non-rigid-body, which for our parameters has d xy symmetry.”
The authors state this as an assumption.
Two bands n=1,2 are filled.
“Assuming two bands n = 1 , 2 are filled, one can use the non-abelian Berry phase”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Venues for bulk measurements of the topological characteristics of classical systems have been suggested in photonic systems.
“Nevertheless, venues for bulk measurements of the topological characteristics of classical systems have been suggested in photonic systems 24 .”
From Topolectrical-circuit realization of topological corner modes (2018) · cites this paper as [24]
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Flagged first, then by citations
Research works printed but not linked: 1
- Imhof, S. et al. Topoelectrical circuit realization of topological corner modes. Preprint at https://arxiv.org/abs/1708.03647 (2017) · arXiv
Cited by
Sorted by citations
Topolectrical-circuit realization of topological corner modes
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Designing perturbative metamaterials from discrete models
Matlack, K. H., Serra-Garcia, M., Palermo, A., Huber, S. D., & Daraio, C. (2018) · Nature Materials · doi:10.1038/s41563-017-0003-3 · Cited by 204 · see citing works
Self-citationTopoelectrical circuit octupole insulator with topologically protected corner states
Bao, J., Zou, D., Zhang, W., He, W., Sun, H., & Zhang, X. (2019) · Physical Review B · doi:10.1103/physrevb.100.201406 · Cited by 149 · see citing works
“The first prediction of a second-order topological insulator, based on quantized quadrupole polarization, was demonstrated in classical mechanical [18] and microwave [19] systems, as well as in electrical circuits [20, 21].”
Background
- Authors
- Serra-Garcia, M., Peri, V., Süsstrunk, R., Bilal, O. R., Larsen, T., Villanueva, L. G., & Huber, S. D.
- Affiliations
- ETH Zurich (CH)
- California Institute of Technology (US)
- École Polytechnique Fédérale de Lausanne (CH)
- Corresponding author
- Huber, Sebastian D.
- Venue
- Nature
- Published
- 2018-01-15
- Volume
- 555
- Issue
- 7696
- Pages
- 342-345
- ISSN
- 0028-0836 (ISSN-L)
- 1476-4687
- OpenAlex
- W2746007985
- Semantic Scholar
- 88dbc36d7ab66ce99514be111021d76a0d9feec2
- Cited by
- 982
- References
- 32
- Citations by year
- 2026: 52 · 2025: 96 · 2024: 97 · 2023: 155 · 2022: 121 · 2021: 170 · 2020: 147 · 2019: 100 · 2018: 43 · 2017: 1
- FWCI
- 53.16
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Nature (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CaltechAUTHORS (California Institute of Technology) (submittedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- Infoscience (Ecole Polytechnique Fédérale de Lausanne) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Topological Materials and Phenomena
- Physical Sciences › Engineering › Biomedical Engineering › Acoustic Wave Phenomena Research
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Advanced Condensed Matter Physics
- Keywords
- higher-order topological insulators, mechanical metamaterials, corner states, edge states, Berry phase, multipole moments, phononic crystals
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-06.
Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-06.
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