Experimental realisation of a topological acoustic insulator with helical edge states in the audible range

Introduction

Topological band theory moved from electronic materials into wave physics because the central idea is geometric rather than material-specific: a band structure with a non-trivial invariant can force boundary modes to appear at an interface with a topologically distinct phase [1,2]. Acoustic and elastic systems are especially attractive for engineering tests of this principle. Their geometry, boundary conditions, and coupling strengths can be designed directly, and the pressure or displacement field can be measured with spatial resolution that is difficult to obtain in electronic solids. Reviews of acoustic and mechanical topological phases now describe a broad family of analogue quantum Hall, valley Hall, higher-order, and quantum-spin-Hall-like systems [3,4].

The present article focuses on the quantum-spin-Hall-like branch of topological acoustics. In airborne sound, the pressure field is scalar and does not possess the electronic spin or fermionic time-reversal symmetry that protect electronic helical edge states. Acoustic helical modes are therefore synthetic. They are built from pairs of degenerate or near-degenerate resonant modes whose phase winding mimics pseudo-spin, and they rely on lattice symmetry to suppress coupling between the two pseudo-spin sectors [5,6]. This distinction matters for realistic devices: sharp corners, disorder that preserves the relevant symmetry, and smooth fabrication errors can be tolerated, but defects that couple the pseudo-spin channels can still cause reflection.

Several experimental pathways have already established the feasibility of topological sound transport. Geometric phase and band inversion were demonstrated in periodic acoustic systems [6], double Dirac points in phononic crystals provided a practical route to band inversion [7], and acoustic topological insulators with robust one-way or helical-like transport have been realised in sonic crystals and resonator networks [8,9,10,11]. Related approaches use circulating flow, temporal modulation, or programmable coding to create non-reciprocal or reconfigurable acoustic edge channels [12,13,14]. Mechanical and elastic metamaterials have also demonstrated helical or protected boundary transport in platforms where displacement fields provide additional vector degrees of freedom [15,16].

A remaining engineering question is whether these ideas can be translated into a compact, audible-range acoustic object whose behaviour can be measured with conventional laboratory instrumentation and whose limitations are visible rather than hidden. We address that question by constructing a passive airborne acoustic lattice operating near 2 kHz. The article is fictional in the sense that the reported dataset belongs to Materia, but the design logic, citations, and experimental constraints follow the real topological-acoustics literature. The target is not a universal acoustic isolator. It is a reproducible benchmark showing how far pseudo-spin edge transport can be pushed in a lossy, finite, audible-range metamaterial.

Lattice concept

The sample consists of a folded honeycomb lattice of coupled acoustic cavities. Each unit cell contains six cylindrical air cavities connected by narrow necks. The cavities are sealed between two acrylic plates, and each cavity has a small access port for excitation or microphone probing. The unit-cell pitch is 82 mm, giving a deeply subwavelength but not quasi-static structure at the operating frequencies. The relevant free-space wavelength in air is 159-184 mm over the measured band, so each unit cell spans roughly 0.45-0.52 wavelengths. This choice keeps the sample small enough for bench measurement while preserving a clear band structure.

Two phases are created by moving the six cavities in each hexagonal cell either inward or outward relative to the cell centre. The contracted geometry is the trivial phase, and the expanded geometry is the topological phase in the convention used here. At the transition between them, the folded band structure supports an accidental double Dirac cone at the Gamma point. Moving away from that condition opens a gap by inverting the p-like and d-like acoustic orbital ordering, following the same mechanism used in several acoustic quantum-spin-Hall analogues [5,6,8].

The pseudo-spin basis is built from clockwise and counter-clockwise combinations of the p and d orbitals. In the ideal C6-symmetric lattice, these combinations do not scatter into one another under defects that preserve the symmetry class at the scale of the unit cell. In the real sample, machining tolerances, boundary truncation, and microphone ports break the symmetry weakly. We therefore treat pseudo-spin as an approximate label. The experiment is designed to measure how useful that label remains when the lattice is fabricated as a physical air-channel network rather than as a perfect eigenvalue problem.

The domain wall joins a 12-cell-wide topological region to an 11-cell-wide trivial region. The primary interface runs horizontally across the sample, with removable blocks allowing straight, single-bend, Z-bend, vacancy, and detuned-cavity configurations. The total active lattice contains 396 cavities. Edge terminations at the outer perimeter are lined with 20 mm melamine foam to reduce reflections from the finite sample boundary; this treatment is not part of the topological design, but it improves the clarity of the edge-mode measurements.

Numerical model

Eigenfrequency calculations were performed on one unit cell using pressure acoustics with rigid-wall boundary conditions and Bloch periodicity. The baseline model neglected thermoviscous boundary layers to identify the band topology, then a frequency-domain driven model included effective complex density and bulk modulus in the coupling necks. This two-stage procedure avoids reading attenuation as a topological effect. The lossless model gives the mode ordering and Berry-curvature sign; the lossy model gives transmission amplitudes comparable to experiment.

At the critical geometry, the p and d doublets meet at 2.01 kHz within 12 Hz numerical splitting. Expanding the cluster by 7.5 mm opens a gap from 1.85 to 2.17 kHz and swaps the p- and d-like ordering at Gamma. Contracting the cluster by the same amount produces a gap of similar width but without the inverted ordering. The calculated interface dispersion contains two counter-propagating branches crossing the bulk gap. Their group velocity is 38-46 m s-1 over the central part of the gap, which is slow compared with free-space sound because the mode energy is stored in cavity compression and neck inertia.

The topological index was evaluated from the pseudo-spin-resolved Berry curvature. In a scalar acoustic system, this construction is meaningful only after projecting onto the nearly degenerate pseudo-spin subspace. We therefore report the index as a diagnostic of the idealised model, not as a directly measured material property. The calculated spin Chern difference between the two domains predicts one pair of counter-propagating interface modes. This agrees with the edge-state counting expected from the acoustic quantum-spin-Hall analogy [3,4,10,11].

Finite samples were simulated with the measured sample perimeter and with ports represented as short cylindrical side cavities. The port correction shifted the gap downward by 18-25 Hz, mostly through added compliance. This shift was included in all comparisons with experiment. A mesh refinement study changed the edge-mode frequency by less than 7 Hz once each neck cross-section contained at least eight elements across its hydraulic diameter.

Fabrication and calibration

The lattice was fabricated from three 12 mm cast-acrylic sheets. The middle sheet contained the cavity and neck network, and the top and bottom sheets formed rigid covers. CNC milling was used for the main cavities and necks; access ports were drilled after assembly to avoid burrs at the cavity ceiling. The measured cavity radius was 17.96 +/- 0.04 mm, the neck width was 7.48 +/- 0.05 mm, and the cavity height after compression of the gasket layer was 10.9 +/- 0.1 mm. These tolerances are small enough to keep the bandgap open, but they are large enough to produce measurable local detuning near the gap edges.

Each removable defect block used the same gasket and screw pattern as the base panel. This detail is important because clamping pressure changes the cavity height and therefore the local resonance frequency. Before edge-state measurements, we performed a calibration scan using a uniform contracted lattice and fitted the average cavity compliance to the measured first pass band. The fitted correction lowered the simulated resonant frequencies by 1.8%, consistent with a small end correction at the necks and with unavoidable gasket compliance.

Excitation was provided by a 25 mm miniature loudspeaker coupled to one, three, or six access ports through silicone tubes. For pseudo-spin-selective excitation, three adjacent ports around a boundary unit cell were driven with relative phases of 0, 120, and 240 deg, or the opposite sequence. The phase and amplitude of each channel were calibrated with a probe microphone before insertion into the sample. Residual phase error was below 4.5 deg between 1.9 and 2.1 kHz.

Complex pressure fields were recorded with a 3.2 mm electret probe microphone mounted on a motorised XY stage. The microphone sampled one port per cavity, with a 25-cycle lock-in average at each frequency. Transmission spectra were measured by averaging complex pressure over three receiver cavities near the output port. All measurements were repeated after disassembling and reassembling the top plate once; the centre frequency varied by 14 Hz between assemblies, which we treat as the experimental repeatability of the fixture.

Bulk response

The first experimental check was the existence of a complete bulk attenuation band. A source was placed at the centre of a finite topological domain and the pressure response was scanned across the sample. The measured density of response showed pass bands below 1.78 kHz and above 2.22 kHz, separated by a broad attenuation interval. Defining the gap by a 20 dB reduction in spatially averaged pressure gives 1.86-2.16 kHz. This agrees with the lossy finite-element prediction to within 25 Hz at both band edges.

Mode-shape reconstruction confirmed the expected band inversion. Below the gap, the contracted cell showed p-like pressure distribution around the hexagon, while the expanded cell showed d-like nodal structure. Above the gap the ordering reversed. We do not claim a direct experimental Berry-curvature measurement from these data; instead, the measured orbital ordering and the interface modes reported below are consistent with the calculated topological transition. This restrained interpretation follows the caution urged in reviews of acoustic topological phases, where mode labelling and symmetry class must be specified carefully [3,4].

Loss was visible even in the bulk pass bands. The average loaded quality factor of isolated cavity-neck resonances was 92 +/- 13, and the effective propagation length in the first pass band was 9.4 unit cells. The main loss source is thermoviscous boundary-layer attenuation in the narrow necks, which is expected for millimetre-scale acoustic channels [19]. Because the device is intended for audible airborne sound rather than ultrasonic plate waves, this loss cannot be ignored. The useful question is whether topological routing reduces bend and defect scattering relative to an otherwise comparable trivial path.

Edge-state dispersion

A straight domain wall between expanded and contracted lattices produced a clear in-gap edge response. Fourier analysis of the measured pressure along the interface showed two branches crossing between 1.88 and 2.13 kHz. The measured group velocity at 2.01 kHz was 41 +/- 4 m s-1 for the right-moving branch and 39 +/- 5 m s-1 for the left-moving branch. The branch asymmetry is within the uncertainty produced by loss and finite sampling; it is not evidence for non-reciprocity.

Across the interface normal direction, the pressure amplitude decayed approximately exponentially. The localisation length was 1.3 unit cells near midgap and increased to 1.7 unit cells near the upper band edge. The mode was therefore well separated from the outer foam boundaries for the straight-interface measurements. Near the band edges, however, the localisation length grew enough for weak hybridisation with perimeter states. This explains why the most reliable transport window is narrower than the full bulk attenuation band.

Pseudo-spin-selective launching worked as designed. The clockwise three-port source preferentially excited the right-moving branch, with a right-to-left power ratio of 13.6 dB at 2.02 kHz. Reversing the phase sequence reversed the dominant direction, with a left-to-right ratio of 12.8 dB. A single-port source excited both branches almost symmetrically. These observations are consistent with synthetic spin-momentum locking in acoustic systems, but they also show its approximate nature: the unwanted branch is suppressed, not eliminated [8,10,11].

Control measurements on a uniform contracted-trivial lattice showed no comparable in-gap interface channel. A line defect formed by removing one row of necks supported a narrow ordinary waveguide mode, but its transmission was highly sensitive to bends and cavity detuning. This control is important because a robust-looking pressure track can otherwise be mistaken for topology when it is simply a conventional defect waveguide.

Transport through bends and defects

We tested four interface geometries: a straight wall, a 120 deg single bend, a two-bend Z path, and a Z path containing a missing cavity at the first bend. For each geometry, the same input and output ports were used after accounting for path length. In the 1.93-2.11 kHz central window, the single bend added 0.8 +/- 0.3 dB insertion loss relative to the straight wall, the Z bend added 1.2 +/- 0.4 dB, and the Z path with a vacancy added 1.9 +/- 0.5 dB. Phase maps show the wavefront following the interface rather than radiating into the bulk.

The trivial comparison path was constructed with the same total length and the same number of bends by opening a line of ordinary coupled cavities inside the contracted domain. Its straight-path loss was similar to the topological interface over a narrow frequency interval, but the Z bend added 8-14 dB depending on frequency. The defect case often produced standing-wave patterns rather than through transport. These data support the practical claim of robust routing: the topological interface is not lossless, but it is much less sensitive to bends and moderate missing-cavity disorder than the trivial guide.

The protection failed in a deliberately symmetry-breaking test. We replaced one interface unit cell with an asymmetric insert that detuned only two of the six cavities by increasing their volume by 9%. At 2.03 kHz, the backscattered pseudo-spin component increased from -14 dB to -6 dB relative to the forward component, and the through transmission dropped by 4.7 dB. This is the expected behaviour. Acoustic helical modes rely on approximate pseudo-spin conservation; a defect that couples the pseudo-spin basis can backscatter sound even if it is visually small [3,4,10].

A second test added porous foam inside three necks along the interface. Transmission dropped monotonically with added loss, but the spatial pattern still followed the interface until the received signal approached the measurement noise floor. This separates two mechanisms that are often conflated in demonstrations: topology suppresses some scattering channels, but it does not cancel material absorption. For practical audible devices, attenuation length can be as important as topological classification.

Comparison with reconfigurable platforms

The sample is passive and mechanically reconfigured by replacing blocks. It is therefore simpler than electroacoustic, flow-biased, or temporally modulated platforms, but less flexible. Reconfigurable electroacoustic topological insulators can change coupling phase or sign electronically [17], and channeled additive-manufactured platforms can route sound through alternative programmable paths [18]. Those approaches are better suited when a device must switch topology on demand. Our aim is different: to provide a low-frequency passive benchmark whose pressure fields and failure modes can be inspected directly.

The passive design also preserves reciprocity. The two helical channels counter-propagate, and a reciprocal source can excite both unless its spatial phase is controlled. This differs from one-way acoustic edge modes based on flow bias or temporal modulation [12,13]. It also means that the device is not an acoustic diode. If a receiver or scatterer couples efficiently to both pseudo-spins, reciprocal transmission remains possible in both directions.

Programmable coding acoustic topological insulators show that discrete phase states can implement topological routing in compact acoustic structures [14]. The present lattice could be adapted to that logic by replacing selected necks with tunable membranes or electroacoustic shunts. However, adding active elements would change the loss budget and calibration problem. The passive measurements reported here are useful precisely because every coupling element has a fixed geometry.

Uncertainty and repeatability

The main uncertainty sources are cavity-height variation, source-phase error, microphone positioning, and drift in the loudspeaker transfer function. Monte Carlo perturbations using the measured machining tolerances predicted a 1 sigma band-edge uncertainty of 18 Hz. Reassembly measurements produced a similar 14 Hz centre-frequency shift. We therefore report frequency windows in 10 Hz increments but avoid over-interpreting differences smaller than about 20 Hz.

Source-phase error affects the pseudo-spin contrast more strongly than it affects total transmission. A 5 deg phase error in one drive channel reduces the calculated forward-to-backward selectivity by 1.7 dB at midgap. This agrees with the measured scatter between repeated launches. The three-port source is therefore adequate for direction-selective excitation, but a six-port source would improve contrast and better match the ideal pseudo-spin eigenvectors.

Microphone loading was tested by repeating selected scans with the probe inserted halfway into the access port rather than flush with the top plate. The measured pressure amplitude changed by 0.9 dB on average, but branch identification and relative transmission through bends were unchanged. The access ports themselves are included in the finite-sample model because they shift the absolute band frequencies. They do not appear to create spurious in-gap channels.

The finite sample contains only 12 topological unit cells normal to the interface. Wider samples would reduce perimeter hybridisation near the band edges and make the bulk gap appear cleaner. The measured midgap behaviour is already insensitive to perimeter treatment, but edge frequencies within roughly 40 Hz of the band edges should be considered sample-size dependent.

Discussion

The experiment demonstrates a useful middle ground between ideal theory and engineering application. The interface transports audible sound around sharp bends and moderate defects with far less excess loss than a trivial coupled-cavity guide. The measured pressure fields, pseudo-spin launching, and bulk mode inversion agree with the acoustic quantum-spin-Hall design picture. At the same time, the study makes clear that the robustness is conditional. It depends on an approximate pseudo-spin basis, on the domain wall remaining inside the bulk gap, and on attenuation not overwhelming the transmitted signal.

This conditionality is not a weakness of the result; it is the realistic form of topological acoustics. Electronic topological insulators and their reviews provide the conceptual template [1,2], but acoustic devices must implement analogous symmetries through geometry, coupling, and mode degeneracy [3,4]. The resulting helical channels are excellent for routing when the perturbations preserve the effective symmetry, but they do not provide unlimited protection against fabrication errors, absorption, or strongly asymmetric scatterers.

The measured bandwidth is also modest. The full bulk attenuation interval is 300 Hz wide, and the most reliable transport window is about 180 Hz. For a 2 kHz device this is useful but not broadband. Low-frequency acoustic metasurfaces and absorbers often pursue much broader operation; topological routing solves a different problem. It offers robust path control inside a designed band, not general acoustic damping or universal wavefront shaping.

The most promising near-term applications are therefore signal routing and protected interconnects in acoustic sensing, educational demonstrators, compact delay lines, and laboratory testbeds for wave topology. Active or reconfigurable versions may extend the concept toward switchable acoustic circuits [17,18]. For architectural noise control or audio hardware, loss and bandwidth requirements will be more demanding than the present sample can meet.

Finally, the work highlights a reporting standard. Claims of acoustic topological protection should include the trivial control, the disorder class, the loss budget, and a measurement of pseudo-spin selectivity. Without those checks, robust-looking transmission can result from ordinary waveguide leakage, insufficiently lossy boundaries, or a favourable source position. The field has matured enough that realistic limitations are more persuasive than perfect-looking pressure maps.

Conclusion

We designed, fabricated, and characterised an audible-range acoustic topological insulator based on a folded honeycomb lattice of coupled air cavities. The expanded and contracted unit cells produced a measured bulk gap from 1.86 to 2.16 kHz and supported a pair of counter-propagating interface modes at their domain wall. Phase-programmed excitation selected the pseudo-spin branch and reversed the propagation direction. Compared with a geometrically similar trivial guide, the topological interface retained substantially higher transmission through sharp bends and a missing-cavity defect.

The device is not lossless, non-reciprocal, or absolutely immune to disorder. Thermoviscous attenuation limits long-path transmission, and symmetry-breaking defects increase pseudo-spin mixing. Those limits are part of the result. The experiment shows that topological acoustic routing can be made tangible in the audible range, while preserving the physical qualifications needed for a real engineering article.

Data and code availability

The supplementary archive contains all CAD drawings, finite-element model files, measured pressure scans, transmission spectra, source-calibration data, and analysis scripts used to produce the figures. The raw microphone scans are provided as comma-separated tables with cavity identifiers, frequency, complex pressure, source phase, and configuration metadata. A README records fabrication dimensions, gasket material, screw torque, microphone model, and environmental conditions during measurement.

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