Flatbands from bound states in the continuum for orbital angular momentum localization

Domain: chemistry
Published: 2026-02-23
Editorial outcome: minor_revision

Abstract
Flatband materials exhibit energy bands with zero dispersion, allowing wavefunctions to be compactly localized on the scale of a unit cell. However, such compact localization does not generally apply to wavefunctions with complex structures, such as those carrying orbital angular momentum (OAM). This limitation arises from the fact that a Bloch wavefunction consists of a plane wave factor e^{ik\cdot r} multiplied by a periodic function u(r). While a flatband flattens the dispersion of the plane wave factor, the compact localization of a general wavefunction additionally requires a highly degenerate periodic function to accommodate its internal structure. Here, we introduce a general framework for constructing such highly degenerate flatbands by leveraging bound states in the continuum (BICs). We experimentally demonstrate this framework in two- and three-dimensional (2D and 3D) acoustic crystals, realizing flatbands with four-fold and twelve-fold degeneracy, respectively. The resulting internal degrees of freedom enable the compact localization of complex structured fields with OAM in both 2D and 3D. Our results not only establish a viable platform for OAM-compatible flatband filtering for acoustic signal processing, but also open new avenues for the construction of topologically structured waves.

Introduction
Flatband materials, characterized by dispersionless energy bands and Bloch wavefunctions, support compactly localized states that arise not from defects or disorder, but from intrinsic destructive interference in the scale of a unit cell in a lattice. These flatband modes, residing in the bulk rather than at certain boundaries, have been realized across various lattice geometries, such as twisted bilayer graphene, kagome lattice and Lieb lattice. The resulting compact localization strongly enhances inter-particle correlations and wave-matter interactions, giving rise to exotic phenomena such as fractional Chern insulators, and enabling applications in distortion-free storage, quantum state transfer and reconfigurable laser arrays.

Despite this remarkable progress, a fundamental question remains: What kinds of wavefunctions can be compactly localized by a flatband? In fact, we notice that so far compact localization does not generally apply to wavefunctions with complex structures, such as those carrying OAM. Unlike amplitude, phase, and polarization, OAM offers a spatially structured degree of freedom, characterized by quantized topological charge and chirality of phase fronts. It has transformed fields including optical communication, quantum information, structured wave control, and light-matter interaction. However, its integration with flatband platforms has never been realized, and no general framework currently exists for the compact localization or manipulation of OAM in flatband systems.

The reason is because the flatness of the energy band alone is insufficient for such integration. In a crystal environment, it is well-known that a Bloch wavefunction takes the general form of u(r)e^{ik\cdot r}, where e^{ik\cdot r} is a plane wave with momentum k that carries long-range phase information across the lattice, and u(r) is a function with the same periodicity as the lattice, describing how the wavefunction behaves within a unit cell. In the scenario of a flatband, the flatness, or momentum independence of the energy spectrum, provides sufficiently many k values to interfere destructively, leading to compact localization on the scale of a unit cell. However, it is the function u(r) that determines the internal structure of the wavefunction. Therefore, to accommodate complex internal structures like those of OAM, the function u(r) must possess high degeneracy to provide sufficient degrees of freedom.

Achieving this goal requires a new class of flatband materials with high modal degeneracy. Yet most existing flatband platforms – whether electronic, photonic, or phononic – are based on non-degenerate bands and offer limited modal diversity. Some theoretical studies have proposed degenerate flatbands, but these often rely on discrete tight-binding models and are typically limited to 2D geometries. As for another type of angular momentum, spin, which is an intrinsic quantum property of electrons, a recent theory has proposed flat-band states with “stiff spin chirality”, but these states do not achieve compact localization of spin and have not been experimentally realized. A scalable and physically intuitive method for realizing OAM-compatible flatbands across arbitrary dimensions and platforms remains an open challenge.

In this work, we propose a general framework for constructing multifold degenerate flatbands in systems of arbitrary dimensionality, based on bound states in the continuum (BICs) – non-radiative modes that reside within the continuous spectrum. These BICs are embedded within individual unit structures and are transformed into flatbands when the units are periodically arranged into a lattice. The unit structures hosting BICs can originate from any type of open wave system, and the lattice can possess any spatial symmetry, independent of its dimensionality. This approach goes beyond tight-binding models, providing a universal and scalable method for realizing highly degenerate flatbands across diverse physical platforms. We demonstrate the concept using two acoustic crystal models in two and three dimensions, featuring flatbands with four-fold and twelve-fold degeneracy, respectively, which enable the compact localization of OAM in 2D and 3D. Our work not only establishes a practical platform for OAM-compatible flatband filtering for acoustic signal processing, but also opens new avenues for the construction of topologically structured waves.

Results
General construction
Our starting point is an open single unit that supports several radiative modes and one or more non-radiative modes, namely, BICs. Owing to the open nature of the structure, we construct a lattice by periodically arranging the single unit and connecting neighboring units through their open boundaries. In this configuration, the original radiative modes in each unit hybridize with those in adjacent units, forming dispersive Bloch bands. In contrast, the BICs, due to their non-radiative character, remain confined within individual unit cells and do not couple to other modes, either in the same unit or across neighboring units. As a result, these BICs are transformed into flatbands. This intuitive picture shows that the properties of the resulting flatbands are entirely determined by the single unit and its BICs, greatly simplifying the design of flatband systems, particularly in higher dimensions or with high degeneracy.

The geometry of the single unit determines how it can be connected to form a lattice, thereby controlling the symmetry and dimensionality of the resulting system. As we demonstrate later, by tailoring the geometry of the unit, flatband systems can be realized in different dimensions. Moreover, since every BIC in the unit becomes a flatband in the lattice, the mode profiles and degeneracy of the flatbands exactly mirror those of the original BICs. This one-to-one correspondence enables precise control over both the existence and the detailed structure of flatbands.

Waveguide network model
Building on this framework, we introduce a waveguide network model to realize various flatband lattices. The basic unit, illustrated in Fig. 1a, consists of two resonant waveguides and one transport waveguide. Open boundary conditions are applied at both ends of the transport waveguide, allowing radiation into free space. The resonant waveguides act as cavities, each resonating at a frequency determined by v/\lambda_0, where v is the wave speed, and \lambda_0 is the free-space wavelength. These two resonant waveguides interact through the propagating field in the transport waveguide.

Due to the open boundaries, the eigenmodes of this structure are generally lossy, with BICs being the notable exceptions. Specifically, a BIC can be realized at the resonant frequency v/\lambda_0 by setting the distance between the two resonant waveguides to \lambda_0/2. The eigenmode characteristics can be understood through the following theoretical analysis. At the resonant frequency v/\lambda_0, the field distribution in both the resonant and transport waveguides forms a standing-wave pattern, ensuring that the field amplitude at the junctions is zero. The eigenmode can be expressed as a superposition state |\psi_{BIC}\rangle = p_1 |p_1\rangle + p_2 |p_2\rangle + m_1 |m_1\rangle, where |p_1\rangle and |p_2\rangle represent the waveforms in the two resonant waveguides, and |m_1\rangle corresponds to the field in the transport waveguide between them. The coefficients p_1, p_2 and m_1 must satisfy the velocity conservation at the two junctions: p_1 + m_1 = 0 and p_2 + m_1 = 0. This leads to the eigenmode |\psi_{BIC}\rangle = m_1 ( -|p_1\rangle - |p_2\rangle + |m_1\rangle ), which is a BIC. When this single unit is used as a unit cell of a 1D lattice, it gives rise to a flatband at the frequency v/\lambda_0 (see “Methods” and Sections I-II of Supplementary Information).

The dimensionality and degeneracy of flatbands can be increased by modifying the design of the unit cell. As an example, consider the structure shown in Fig. 1b, obtained by performing a four-fold rotation operation to the original unit. The eigenmode at the same resonant frequency v/\lambda_0 can now be written as |\psi_{4BIC}\rangle = p_1 |p_1\rangle + p_2 |p_2\rangle + p_3 |p_3\rangle + p_4 |p_4\rangle + m_1 |m_1\rangle + m_2 |m_2\rangle + m_3 |m_3\rangle + m_4 |m_4\rangle, involving eight degrees of freedom. Velocity flow conservation at the four junctions leads to four constraints: p_1 + m_1 + m_4 = 0, p_2 + m_1 + m_2 = 0, p_3 + m_2 + m_3 = 0 and p_4 + m_3 + m_4 = 0. This leaves four independent coefficients, which we choose to be m_1, m_2, m_3, and m_4. The resulting eigenmode becomes superposition of four linearly independent states, each forming a BIC: |\psi_{4BIC}\rangle = m_1 (-|p_1\rangle - |p_2\rangle + |m_1\rangle) + m_2 (-|p_2\rangle - |p_3\rangle + |m_2\rangle) + m_3 (-|p_3\rangle - |p_4\rangle + |m_3\rangle) + m_4 (-|p_4\rangle - |p_1\rangle + |m_4\rangle). Thus, the single unit supports a four-fold degenerate set of BICs at frequency v/\lambda_0. This unit can be arranged into a 2D lattice (Fig. 1c), resulting in a four-fold flatband. Later, we show that such degenerate BICs can be used for compact localization of OAM.

In general, the degeneracy of the eigenmodes can be expressed as N = N_w - N_n, where N_w is the number of waveguides and N_n is the number of nodes (junctions). For the 2D structure, N_w = 8 and N_n = 4. Following this principle, one can design unit cells with various combinations of N_w and N_n, including those that support 3D lattice construction. As an extreme case, Fig. 1d shows a unit containing 12 transport waveguides, 8 resonant waveguides and 8 nodes, supporting 12-fold degenerate BICs. When arranged into a 3D lattice (Fig. 1e), this unit gives rise to a 12-fold flatband. Importantly, the lattice geometry is not restricted to square or cubic arrangements and can be extended to other types like the triangle lattice (see Sections III-IV of Supplementary Information).

Experimental realization of four-fold flatbands in a 2D acoustic crystal
Next, we consider a realistic implementation of the BIC concept. The design consists of two resonant waveguides and one transport waveguide, as shown in Fig. 1a. The main difference from the theoretical model is the inclusion of waveguide diameter. We fix the diameter of all waveguides to d = 8 mm and the height of the resonant waveguides to h = 20 mm, while treating the distance l between two resonant waveguides as a tunable parameter. Theory predicts that a BIC forms when l = 2h. However, due to the finite cross-section of the waveguides, this condition is slightly modified in practice. To determine the actual BIC condition, we calculate the eigenfrequencies for different values of l using finite-element simulations (see “Methods” for numerical details). A BIC with a purely real eigenfrequency is found when l = 36 mm (see Section V of Supplementary Information). The BIC occurs at around 5 kHz, differing from the theoretical prediction of v/(4h) \approx 4.288 kHz, due to the fact that the diameter of the waveguides is comparable to the height h.

The acoustic waveguide system serves as a fundamental building block for constructing higher-dimensional composite structures that support degenerate BICs. These can be used as unit cells for acoustic crystals exhibiting multifold flatbands. Figure 2a shows an example: a 2D acoustic crystal arranged in a square lattice. The entire structure is air-filled and bounded by photosensitive resins, which act as rigid walls for sound. Each unit cell contains four resonators and four transport waveguides, with a lattice constant of a_s = 54 mm. As predicted by our theory, each unit cell hosts four-fold degenerate BICs, leading to a four-fold degenerate flatband in the 2D lattice.

We measure the acoustic pressure field excited by a single source placed at the center of the sample (see “Methods” for experimental details). By applying a Fourier transform, we extract the band structure of the system. The experimentally measured and numerically simulated dispersions are shown in Fig. 2b, c, respectively, both clearly revealing flatbands around 5 kHz. The right panel of Fig. 2c shows the eigenfields of the four flatband modes at the \Gamma point. Due to the non-zero cross-section area of the waveguides, the flatband frequencies are slightly split. Interestingly, eigenstates f_1 and f_4 preserve the C_{4v} symmetry of the unit cell, while f_2 and f_3 break this symmetry. Thus, f_2 and f_3 must form a degenerate pair to comply with the system’s symmetry requirement. f_4 deviates largely from f_2 and f_3 for their field distribution, the deviation is 261 Hz. f_1 deviates less, the deviation is 106 Hz. Note that the highest-frequency flatband mode (f_4) is not observed in the experiment, as it has negligible field amplitude in the resonant waveguides, from which signals are measured (see eigenmode profiles in Fig. 2c).

Compact localization of OAM
While compact localization can be achieved with non-degenerate flatbands, multifold flatbands offer greater tunability in the spatial profile due to their high degeneracy. As predicted, the eigenmode of localized states in the 2D model can be written as |\psi_{4BIC}\rangle = m_1 (-|p_1\rangle - |p_2\rangle + |m_1\rangle) + m_2 (-|p_2\rangle - |p_3\rangle + |m_2\rangle) + m_3 (-|p_3\rangle - |p_4\rangle + |m_3\rangle) + m_4 (-|p_4\rangle - |p_1\rangle + |m_4\rangle), where m_1, m_2, m_3, and m_4 are adjustable coefficients. Different choices of these values produce different localized states. For example, choosing m_1 = m_2 = m_3 = m_4 = +1, the state simplifies to |\psi_{4BIC}\rangle = -2|p_1\rangle - 2|p_2\rangle - 2|p_3\rangle - 2|p_4\rangle + |m_1\rangle + |m_2\rangle + |m_3\rangle + |m_4\rangle.

Such a flatband-localized state can serve as a flatband filter, selectively extracting the flatband frequency from a broadband signal. To demonstrate this, we excite the system using a Gaussian pulse of the form s_i = e^{-\pi^2 f_0^2 t^2} \cos(2\pi f_c t + \phi_i) with bandwidth parameter f_0 = 1000 Hz, center frequency f_c = 5000 Hz, and phase \phi_i (The Gaussian waveform is chosen for its experimental convenience, though other waveforms may also be used). As indicated in the upper panel of Fig. 2d, the pulse is applied to the four transport waveguides, and the response is measured through the four resonant waveguides labeled as “cav.1" to “cav.4", respectively. The acoustic pressures measured at the four neighboring resonators exhibits synchronized temporal oscillations and a resonant peak at approximately 4990 Hz. As a hallmark flatband feature, the fields remain compactly localized without spreading and phase-synchronized over time, even after the excitation has ceased, as confirmed in Fig. 2g and Supplementary Movie I.

While flatband filtering is a natural consequence of flatbands, here we demonstrate that our multifold flatbands can further incorporate OAM information, enabling OAM-compatible flatband filtering. Specifically, by choosing m_1 = +1, m_2 = +i, m_3 = -1 and m_4 = -i, we achieve compact localization of OAM with counterclockwise angular momentum, |\psi_d\rangle = (-1+i)|p_1\rangle + (-1-i)|p_2\rangle + (1-i)|p_3\rangle + (1+i)|p_4\rangle + |m_1\rangle + i|m_2\rangle - |m_3\rangle - i|m_4\rangle. For such a state, the resonance frequency is selectively excited at the flatband frequency, the wavefunction is compactly localized, while the OAM is preserved. This time we use a chiral source for excitation, which is the same Gaussian pulse with its \phi_i shifted by four different values as indicated in the upper panel of Fig. 2e. The fields in the four resonators continue to oscillate with the same period, but the phase curves are offset by equal intervals in the clockwise direction (Fig. 2e). Fourier analysis reveals a single resonant peak around 5156 Hz, similar to the former case. However, the phases at this frequency now increase in the counterclockwise direction (Fig. 2e) and a compact OAM localization is observed (Fig. 2h). Notably, the relative phases among the four resonators remain synchronized, ensuring that the OAM is preserved over time despite overall signal decay due to loss (see Supplementary Movie II).

If the chiral excitation switches the chirality, namely, is changed to m_1 = +1, m_2 = -i, m_3 = -1, and m_4 = +i, we realize compact localization of OAM with clockwise angular momentum: |\psi_d\rangle = (-1-i)|p_1\rangle + (-1+i)|p_2\rangle + (1+i)|p_3\rangle + (1-i)|p_4\rangle + |m_1\rangle - i|m_2\rangle - |m_3\rangle + i|m_4\rangle. Measurements under this rearranged chiral source yield similar results, with the direction of the OAM reversed (Fig. 2f, i). These results align with numerical simulations (see Section VI of Supplementary Information). Furthermore, the compactly localized OAM can be programmed into arbitrary patterns, as each unit cell’s OAM state can be controlled independently. For example, loop states with spatially winding OAM phases can be constructed (see Section VIII of Supplementary Information).

Experimental realization of twelve-fold flatbands in a 3D acoustic crystal
Finally, we demonstrate an extreme case: twelve-fold flatbands in a 3D acoustic crystal (Fig. 1e). A photo of the sample with 10 \times 10 \times 10 unit cells is shown in Fig. 3a. The unit cell design matches our theoretical model, with resonator height h = 17.2 mm, waveguide diameter d = 6 mm, and lattice constant a_s = 40 mm. The lower-right panel of Fig. 3a shows one of the twelve flatband modes, which is compactly localized within the unit cell. Using a point source, we excite the structure and measure fields in the transport waveguides. A Fourier transform yields the band structure, which is shown in Fig. 3b alongside numerical results. Both exhibit flatbands near 6.95 kHz. Numerical simulations confirm the degeneracy is twelve-fold, as expected. This high degeneracy leads to stronger Fourier intensity for the flatband compared to the 1D and 2D cases.

The high modal degeneracy of our system enables compact localization of OAM at arbitrary locations and along any direction. By appropriately configuring the excitation, one can generate nonzero OAM along directions such as (1,0,0), (0,1,0), (0,0,1), or (1,1,1). For example, to excite a localized flatband mode with OAM along (0,0,1) (i.e., the z-axis), we use four sources positioned and phased as illustrated in the lower-right panel of Fig. 3c. This four-source configuration results in a compactly localized OAM mode centered within a single unit cell, as evidenced by the measured amplitude and phase fields in Fig. 3c. A cross-sectional view shown in Fig. 3d reveals that the field is highly confined to the central unit cell, and the phases of the four central resonators exhibit a 2\pi counterclockwise winding — clearly indicating the presence of nonzero OAM along the z direction.

To further demonstrate the versatility of our approach, we realize compact OAM localization along the (1,1,1) direction. The lower-right panel of Fig. 3e illustrates a configuration of twelve sources: four each designated to excite OAM components along the x-, y-, and z-axes (colored with red, blue, and black, respectively). Together, these sources produce a localized OAM mode aligned along the (1,1,1) direction. The measured field amplitude and phase shown in Fig. 3e confirm localization within the central unit cell, with the field primarily concentrated in six resonators distributed across two (1,1,1) planes. Fig. 3f shows the results from one of these planes, where a 2\pi counterclockwise phase winding is observed, unambiguously confirming the presence of OAM along the (1,1,1) direction.

Topologically structured acoustic fields in a 3D lattice
Recently, 3D topologically structured optical fields have been demonstrated in photonics, offering promising prospects for future information carriers, including qubits. Building upon our ability to achieve compact OAM localization in arbitrary directions, we demonstrate that these localized OAM modes can serve as fundamental elements for constructing 3D topologically structured fields for sound. This provides a practical strategy for temporally storing such topologically structured fields in a compact and localized manner. Figure 4a shows a torus-shaped field configuration characterized by the topological invariant (\mu_1, \mu_2) = (0, 1), where \mu_1 and \mu_2 are two winding numbers defined on closed loops C_1 and C_2, respectively. These winding numbers are calculated using the integral \mu_{1,2} = 1/(2\pi) \oint_{C_{1,2}} d\phi, where \phi = \arg(p) is the phase of the acoustic pressure field. This topological configuration is experimentally realized by exciting eight unit cells with appropriately oriented OAM, as indicated by the golden arrows in Fig. 4a, b. As shown in Fig. 4c, the measured acoustic field exhibits a torus structure. The phase measurements confirm a 2\pi phase winding on the zx-plane, but not on the yz-plane, validating the (0, 1) topological invariant.

We further demonstrate a more complex configuration with a topological invariant (\mu_1, \mu_2) = (1, 1), as illustrated in Fig. 4d. This topological configuration carries a nonzero Hopf invariant Q_{Hopf} = \mu_1 \mu_2 = 1, dictating the presence of a Hopf link formed by two interlinked equiphase curves (shown in blue and red). From the OAM perspective, this configuration corresponds to a continuous phase twist of the OAM around the torus, similar to the scenario discussed in Section VIII of Supplementary Information. This configuration is excited in the time domain using eight unit cells with modified phase inputs. As shown in Fig. 4f, phase windings are clearly observed on both the zx and yz planes, confirming the topological invariant (1, 1).

Methods
Numerical simulations
All simulations are performed using the acoustic module of COMSOL Multiphysics, which is based on the finite element method. The photosensitive resin used for sample fabrication is set as the hard boundary due to its large impedance mismatch with air. The real sound speed at room temperature is c_0 = 343 m/s. The density of air \rho is set to be 1.28 kg/m^3.

Experimental measurements
For the measurement of four- and twelve-fold flatbands, a broadband sound signal (frequency range of 2 Hz – 10 kHz; frequency resolution of 2 Hz) is emitted from a balanced armature sound source placed at the center of the sample. The sound signal is delivered through two narrow tubes (radius of 1.5 mm; lengths of 10 cm and 30 cm). The sound signal is recorded using the software of PULSE Labshop (Brüel & Kjær Type 3160-A-022). For the measurement of localized flatband modes with OAM patterns, a series of sound sources are used to emit the time-domain Gaussian pulse signals with specific phase delays. As an example, to generate signals with phase delays \phi_1 = 0, \phi_2 = \pi/2, \phi_3 = \pi and \phi_4 = 3\pi/2, we first generate two Gaussian pulse excitation signals s_1 (\phi_1 = 0) and s_2 (\phi_2 = \pi/2) using the software of MATLAB. Both signals are exported in standard WAV. audio format and then imported into the software PULSE Labshop. To obtain the remaining two signals s_3 (\phi_3 = \pi) and s_4 (\phi_4 = 3\pi/2), we invert the polarity of the connection between the power amplifier’s output and the sound source excitation terminals, effectively phase-shifting the original signals to \phi_3 = \pi and \phi_4 = 3\pi/2. When driven by these four excitation signals (s_1, s_2, s_3, s_4), the sound sources emit Gaussian pulses with the desired phase delays. By repeatedly launching these signals, we measure and record the time-domain sound response at each detection point. The pressure of each site is detected by a microphone (Bruel & Kjær Type 4961) adhered to a long tube (of diameter about 0.35 cm and a length of 35 cm). The signal is recorded and frequency-resolved by a multi-analyzer system (Bruel & Kjær 3160-A-022 module).

Analytical calculation of dispersion relation for 1D acoustic waveguide system
To analytically calculate the dispersion relation for the 1D acoustic waveguide system in Fig. 1a, we use the general solution of acoustic pressure for the 1D waveguide, that is p(s) = A \cdot e^{iks} + B \cdot e^{-iks}. For a 1D open waveguide, the acoustic particle velocity u_0 at one end of the channel (s = 0) can be represented as a function of pressure p_0, p_L at both ends (s = 0 and L),

u_0 = - \frac{i}{Z_c} \cot(kL) p_0 + \frac{i}{Z_c \sin(kL)} p_L (1)

Where Z_c is the characteristic impedance and k = \omega/v. For a 1D waveguide with hard boundary condition at one end (s = L), the acoustic particle velocity u_0 at the other end of the channel (s = 0) can be represented as a function of pressure p_0,

u_0 = \frac{i}{Z_c} \tan(kL) p_0 (2)

At each node, the interaction point of N waveguides, the acoustic pressure is constrained by the acoustic particle velocity flow conservation law,

\sum_{i=1}^N u_i = 0 (3)

For the 1D model in Fig. 1a, there are two nodes which makes,

\begin{pmatrix} D_{11} & D_{12} \\ D_{21} & D_{22} \end{pmatrix}_{node1, node2} \begin{pmatrix} p_1 \\ p_2 \end{pmatrix} = 0 (4)

with

D_{11} = D_{22} = \tan(k\lambda_0/4) - \cot(k\lambda_0/4) - \cot(k\lambda_0/2)

D_{12} = D_{21}^* = \frac{1 + e^{-iK_B a_s}}{\sin(k\lambda_0/4)} \sin(k\lambda_0/2) (5)

Where K_B is the Bloch momentum. The dispersion relation of the system can be obtained from the determinant of the matrix. After simplification, it is

\cos(K_B a_s) = 1 - \frac{2 \sin(k\lambda_0/4) \sin(k\lambda_0/2) [\tan(k\lambda_0/4) - \cot(k\lambda_0/4) - \cot(k\lambda_0/2)]^2}{\sin^2(k\lambda_0/4) \sin^2(k\lambda_0/2)} (6)

The calculated dispersion relation is shown in Section II of Supplementary Information.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_2": "Thank you for the response.\nI still believe that the flat band constructed by your BIC is trivial, because in your original lattice structure, the formation of the flat band relies on each unit cell being isolated. In your new Hamiltonian, two isolated atoms interact with photons, and although the resulting unit cell is equivalent to your acoustic unit cell, however, when multiple such unit cells are connected, the intra-cell hopping within a unit cell is -iΓe^iπ, while the inter-cell hopping is -iΓe^(iπ/2). This clearly means that the flat band structure no longer exists.", "reviewer_3": "The manuscript combines the idea of Bound States in the Continuum with flat bands in order to localized orbital angular momentum. The approach is simple, interesting, and relatively clear. The authors confirm their design principle numerically and validate it experimentally. The paper has gone through peer review already in Nature Materials and it does seem that the authors have significantly improved the paper compared to the original submission.\nThe remaining two main concerns for my anonymous respected colleague (Referee B) seem to be:\n(1) the \"trivial\" nature of the system as one might be able to cut the system from its surroundings and still obtain \"the same results\".\n(2) the complexity of the experiments and the utility of such material system in \"real world application as memory\".\nFor point (1), I disagree. The measured dispersion curves in Figure 2b show clear dispersive bands. In other words, if one excites this system with a different frequency the waves will propagate normally within the lattice. The waves excited around 4990 Hz show high localization, unlike conventional band gaps where exponential decay would be expected. Figure 3b, does not show a lot of dispersive details due to the high intensity of the localization around the flat bands (although the authors could try to excite the system in segments of frequencies to improve the resolution of bands in the results).\nNevertheless, both figures 2b and 3b demonstrate the fact the system at hand is not really trivial.\nFor point (2), I am not very concerned about the complexity of the excitation, the experiment proves the model, the excitation source can be simplified later [e.g., Nature communications 12, 6954 (2021)]. As for presenting this work as storage or memory, I think the authors need to better present this point as it is not very convincing in the current version of the manuscript and appears to be an afterthought.\nThe clarity of the presentation could be further improved by answering the two questions below:\n(a) How degenerate are the bands? One calls two states degenerate, when they have the same energy level. Here, it would be the degeneracy in frequency: for example f2 and f3 in Figure 2 or f1-f5 in figure 3. However, for bands f1 and f4 in figure 2, it might not be entirely accurate to call these bands degenerate as they are appear to be well separated in frequency. Maybe the authors can quantify that?\n(b) How flat are the bands? [Phys. Rev. Lett. 133, 266101, 2024] quantifies the \"flatness\" of the acoustic flat bands and correlates it to the intensity of localization/transmission. I wonder if the authors could have a similar metric to inform the reader about the flatness of the utilized bands and the localization effectiveness or the wave penetration depth.\n- In figure 2 panels d,e, and f, the left top panel shows the time signal of one of the cavities, while panels g, h, and i show the entire field intensity. What is not very clear is how are the time signals translated into these pillars and later on into circles and spheres. Did the authors take the average of each signal, fft, or max of the time signal?\n- In section VIII in the supplementary materials and in the caption of figure S11 the authors refer to \"32 microphones for excitation\" did the author mean sound sources?\n- The authors should also carefully read the manuscript for typos and grammar:\n(for example) \"cease\" instead of \"crease\" in figure 2 caption, \"negligible\" instead of \"neglectable\", and \"the phase of the acoustic\" instead of \"the phase of he acoustic\"\nIn conclusion, I think this paper is significant and could be of interest to the general readership of Nature Communications after minor revisions."}, "author_response": "Response Letter for the Manuscript “Flatbands from Bound States in the Continuum for Orbital Angular Momentum Localization”\n(NCOMMS-25-90907-T)\nDear Reviewers,\nWe sincerely thank you for your time and valuable insights. Your support and\nsuggestions are invaluable.\nWe thank Reviewer #2 for raising the question regarding whether the new Hamiltonian\nsupports flat bands. We acknowledge this point and would like to clarify that, at the\nresonance frequency, the new Hamiltonian is equivalent to the original model.\nWe thank Reviewer #3 for your positive assessment of our work. We are particularly\ngrateful for your insightful explanation of the concerns raised by Reviewer B, which\nprovides strong support for the non-trivial nature of our findings. In response to your\nother suggestions, we have provided point-by-point replies and have incorporated the\ncorresponding revisions into the revised manuscript.\nIn the text below, the comments from each referee are quoted in italics, followed by our\nresponses.\nBest regards,\nAll the authors\nResponse to Reviewer #2\nReviewer #2 – General comments:\n“I still believe that the flat band constructed by your BIC is trivial, because in your\noriginal lattice structure, the formation of the flat band relies on each unit cell being\nisolated. In your new Hamiltonian, two isolated atoms interact with photons, and\nalthough the resulting unit cell is equivalent to your acoustic unit cell, however, when\nmultiple such unit cells are connected, the intra-cell hopping within a unit cell is\n−𝑖Γ𝑒𝑖𝜋, while the inter-cell hopping is −𝑖Γ𝑒𝑖𝜋/2 . This clearly means that the flat band\nstructure no longer exists.”\nResponse from authors:\nWe thank you for this comment. We believe there may have been a misunderstanding\nregarding the relationship between our original lattice structure and the new\nHamiltonian. In fact, the two models are equivalent to each other near the resonant\nfrequency of the side resonators (in the original lattice structure) or the two-level\nsystems (in the new Hamiltonian). In the original structure, the side resonators interact\nvia acoustic waves in the transport waveguide, which directly corresponds to the\ninteraction between isolated atoms through photons in the waveguide as described by\nthe new Hamiltonian. Thus, the physical mechanisms in both systems are similar.\nIn the new Hamiltonian, each atom couples to all other atoms through far-field photon-\nmediated interactions. Restricting the model to nearest-neighbor coupling alone does\nnot fully capture the system's properties—particularly its band structure. When all long-\nrange couplings are included, a flat band emerges at the resonant frequency, as\ndemonstrated both in the Methods section of the main text under “Analytical calculation\nof dispersion relation for 1D acoustic waveguide system” and in Sec. I of the\nSupplementary Materials titled “Band structure for the 1D acoustic waveguide system\nfrom analytical calculation.” For clarity, we reproduce the relevant band structure from\nthe Supplementary Materials below.\nFig. S1. Band structure for the 1D acoustic waveguide system from analytical\ncalculation.\nResponse to Reviewer #3\nReviewer #2 – General comments:\n“The manuscript combines the idea of Bound States in the Continuum with flat bands\nin order to localized orbital angular momentum. The approach is simple, interesting,\nand relatively clear. The authors confirm their design principle numerically and\nvalidate it experimentally. The paper has gone through peer review already in Nature\nMaterials and it does seem that the authors have significantly improved the paper\ncompared to the original submission.\nThe remaining two main concerns for my anonymous respected colleague (Referee B)\nseem to be:\n(1) the \"trivial\" nature of the system as one might be able to cut the system from its\nsurroundings and still obtain \"the same results\".\n(2) the complexity of the experiments and the utility of such material system in \"real\nworld application as memory\".\nFor point (1), I disagree. The measured dispersion curves in Figure 2b show clear\ndispersive bands. In other words, if one excites this system with a different frequency\nthe waves will propagate normally within the lattice. The waves excited around 4990\nHz show high localization, unlike conventional band gaps where exponential decay\nwould be expected. Figure 3b, does not show a lot of dispersive details due to the high\nintensity of the localization around the flat bands (although the authors could try to\nexcite the system in segments of frequencies to improve the resolution of bands in the\nresults). Nevertheless, both figures 2b and 3b demonstrate the fact the system at hand\nis not really trivial.\nFor point (2), I am not very concerned about the complexity of the excitation, the\nexperiment proves the model, the excitation source can be simplified later [e.g., Nature\ncommunications 12, 6954 (2021)]. As for presenting this work as storage or memory, I\nthink the authors need to better present this point as it is not very convincing in the\ncurrent version of the manuscript and appears to be an afterthought.”\nResponse from authors:\nWe sincerely thank you for your thoughtful evaluation and for your positive assessment\nof our work. We are particularly grateful for your helpful and detailed discussion of the\npoints raised by Referee B.\nRegarding Point (1): The \"Trivial\" Nature of the System\nWe thank you for your helpful analysis and for clearly explaining why the system shows\nnon-trivial wave physics. We completely agree with your assessment. As you rightly\npointed out, the measured dispersion in Figure 2b shows clear propagating bands\noutside the flat-band frequency, confirming that the waveguide lattice stays open and\ndispersive for most frequencies. The strong localization observed at 4990 Hz is different\nfrom conventional band-gap behavior, because it comes from the interference-based\nmechanism of a bound state in the continuum (BIC), not from exponential decay. We\nappreciate your suggestion to possibly improve the resolution of the dispersive bands\nin Figure 3b by using segmented frequency excitation; this is a very good point we will\nconsider for future work. We are glad that you agree the demonstrated phenomena are\nfundamentally non-trivial.\nRegarding Point (2): Experimental Complexity and Utility as Memory\nOn Experimental Complexity: We thank you for your understanding. The main goal of\nthe current experimental setup was to provide a clear and definitive proof-of-concept\nfor the theoretical model. We appreciate your reference to the relevant Nature\nCommunications paper, which shows how such excitations can be made simpler in\nmore applied setups. We agree that practical versions for potential applications would\nfocus on simpler excitation methods.\nOn Presentation as Storage/Memory: We sincerely thank you for your important\nfeedback. We recognize that the discussion of the system's potential for wave storage\nor memory applications in the current manuscript may seem underdeveloped. The main\nfocus of this work is to demonstrate the physical principle—the co-localization of\nenergy and orbital angular momentum through the intersection of a BIC and a flat band.\nWhile the observed long-lived, localized states naturally suggest interesting\npossibilities for temporary wave storage or trapping, we agree that presenting this as a\nmain application needs a more thorough and focused discussion. In the revised\nmanuscript, we have softened the discussion on storage and memory.\nWe have changed the statement “open new avenues for localization and storage of\ntopologically structured waves” to “open new avenues for construction of topologically\nstructured waves”\nReviewer #3 – Comment 1:\n“The clarity of the presentation could be further improved by answering the two\nquestions below:\n(a) How degenerate are the bands? One calls two states degenerate, when they have\nthe same energy level. Here, it would be the degeneracy in frequency: for example 𝑓\n2\nand 𝑓 in Figure 2 or 𝑓 −𝑓 in figure 3. However, for bands 𝑓 and 𝑓 in figure 2,\n3 1 5 1 4\nit might not be entirely accurate to call these bands degenerate as they are appear to\nbe well separated in frequency. Maybe the authors can quantify that?”\nResponse from authors:\nThanks for your constructive suggestion.\nWe first explain why there are band splitting. As our theory predicts, there are 4-fold\ndegenerate flat bands for the 2D model, and the eigenfunctions of flat bands are\n|Ψ ⟩ = 𝑚 (−|𝑝 ⟩−|𝑝 ⟩+|𝑚 ⟩)+𝑚 (−|𝑝 ⟩−|𝑝 ⟩+|𝑚 ⟩)+𝑚 (−|𝑝 ⟩−\n4𝐵𝐼𝐶 1 1 2 1 2 2 3 2 3 3\n|𝑝 ⟩+|𝑚 ⟩)+𝑚 (−|𝑝 ⟩−|𝑝 ⟩+|𝑚 ⟩).\n4 3 4 4 1 4\nIt can be decomposed into four orthogonal eigenstates.\n1. Choosing 𝑚 = 𝑚 = 𝑚 = 𝑚 = 1, we obtain:\n1 2 3 4\n|𝑓 ⟩ = −2|𝑝 ⟩−2|𝑝 ⟩−2|𝑝 ⟩−2|𝑝 ⟩+|𝑚 ⟩+|𝑚 ⟩+|𝑚 ⟩+|𝑚 ⟩\n1 1 2 3 4 1 2 3 4\n2. Choosing 𝑚 = 𝑚 = −𝑚 = −𝑚 = 1, we obtain:\n1 2 3 4\n|𝑓 ⟩ = −2|𝑝 ⟩+2|𝑝 ⟩+|𝑚 ⟩+|𝑚 ⟩−|𝑚 ⟩−|𝑚 ⟩\n2 2 4 1 2 3 4\n3. Choosing 𝑚 = −𝑚 = −𝑚 = 𝑚 = 1, we obtain:\n1 2 3 4\n|𝑓 ⟩ = −2|𝑝 ⟩+2|𝑝 ⟩+|𝑚 ⟩−|𝑚 ⟩−|𝑚 ⟩+|𝑚 ⟩\n3 1 3 1 2 3 4\n4. Choosing 𝑚 = −𝑚 = 𝑚 = −𝑚 = 1, we obtain:\n1 2 3 4\n|𝑓 ⟩ = |𝑚 ⟩−|𝑚 ⟩+|𝑚 ⟩−|𝑚 ⟩\n4 1 2 3 4\nThey correspond to the four states in Fig. 2c.\nIdeally, they should be degenerate. However, in realistic implementations such as\nacoustic crystals, slight frequency splitting is expected due to finite-size effects and\nnonzero cross sections in the actual design. The 𝐶 symmetry of the unit cell requires\n4𝑣\n𝑓 and 𝑓 are degenerate, so |𝑓 ⟩+|𝑓 ⟩ and |𝑓 ⟩−|𝑓 ⟩ are also eigenstates of the\n2 3 2 3 2 3\nsystem, which can be expressed as\n|𝑓 ⟩+|𝑓 ⟩ = 2(−|𝑝 ⟩−|𝑝 ⟩+|𝑚 ⟩)−2(−|𝑝 ⟩−|𝑝 ⟩+|𝑚 ⟩)\n2 3 1 2 1 3 4 3\nand\n|𝑓 ⟩−|𝑓 ⟩ = 2(−|𝑝 ⟩−|𝑝 ⟩+|𝑚 ⟩)−2(−|𝑝 ⟩−|𝑝 ⟩+|𝑚 ⟩)\n2 3 2 3 2 4 1 4\n|𝑓 ⟩ , |𝑓 ⟩+|𝑓 ⟩ , |𝑓 ⟩−|𝑓 ⟩ and |𝑓⟩ are orthogonal to each other and form a\n1 2 3 2 3 4\ncomplete representation to the four flat bands.\nAnalysis of the field distributions of the eigenmodes |𝑓 ⟩, |𝑓 ⟩+|𝑓 ⟩, |𝑓 ⟩−|𝑓 ⟩ and\n1 2 3 2 3\n|𝑓⟩ elucidates the flat band splitting mechanism:\n4\n1. Mode |𝑓 ⟩+|𝑓 ⟩: The field is zero in the x-direction waveguides (|𝑚 ⟩ and |𝑚 ⟩).\n2 3 2 4\nConsequently, these waveguides are unaffected, exhibiting no eigenfrequency\nmodification.\n2. Mode |𝑓 ⟩−|𝑓 ⟩: The field is zero in the y-direction waveguides (|𝑚 ⟩ and |𝑚 ⟩),\n2 3 1 3\nresulting similarly in no eigenfrequency modification.\n3. Mode |𝑓 ⟩ : The field is entirely distributed within the transport waveguides.\n2 4\ndirection waveguides (|𝑚 ⟩ and |𝑚 ⟩) are out of phase. This configuration leads\n1 3\nto the largest eigenfrequency modification.\n4. Mode |𝑓 ⟩: This mode exhibits eigenfrequency modification. However, because the\n1\nfields in the x-direction waveguides and the y-direction waveguides are in phase,\nthe magnitude of this modification is comparatively smaller.\nSo 𝑓 and 𝑓 are degenerate, 𝑓 deviates significantly from 𝑓 and 𝑓 , while f1\n2 3 4 2 3\ndeviates less. The degree of deviation can be directly quantified using frequency\ndifference.\nTo clarify this point, we have added an explanation on Page 3, Right column, which\nreads as: “Due to the nonzero cross-section area of the waveguides, the flatband\nfrequencies are slightly split. Interestingly, eigenstates 𝑓 and 𝑓 preserve the 𝐶\n1 4 4𝑣\nsymmetry of the unit cell, while 𝑓 and 𝑓 break this symmetry. Thus, 𝑓 and 𝑓\n2 3 2 3\nmust form a degenerate pair to comply with the system's symmetry requirement. 𝑓\n4\ndeviates significantly from 𝑓 and 𝑓 for their field distribution, the deviation is 261\n2 3\nHz. 𝑓 deviates less, the deviation is 106 Hz.”\n“(b) How flat are the bands? [Phys. Rev. Lett. 133, 266101, 2024] quantifies the\n\"flatness\" of the acoustic flat bands and correlates it to the intensity of\nlocalization/transmission. I wonder if the authors could have a similar metric to inform\nthe reader about the flatness of the utilized bands and the localization effectiveness or\nthe wave penetration depth.”\nResponse from authors:\nIn our work, the flat bands are formed from bound states in the continuum (BICs). All\nthese flat bands exhibit perfect flatness with strictly zero dispersion, resulting in a\nvanishing bandwidth. These findings are further confirmed by the compact localized\nstates demonstrated in Figs. 2g-2i.\nReviewer #3 – Comment 3:\n“- In figure 2 panels d,e, and f, the left top panel shows the time signal of one of the\ncavities, while panels g, h, and i show the entire field intensity. What is not very clear\nis how are the time signals translated into these pillars and later on into circles and\nspheres. Did the authors take the average of each signal, fft, or max of the time signal?”\nResponse from authors:\nWe thank the reviewer for raising this important point regarding our data processing\nand visualization methodology, which we should have described more clearly in the\noriginal manuscript. Here the data in panels g, h, and i are derived from Fourier\ntransform to time-domain signal.\nTo clarify this point, we have added an explanation in the caption of Fig. 2, which read\nas:” The frequency-domain signals in panels d–f and the field distributions in panels g–\ni are obtained via Fourier transform of the time-domain signals.”.\nReviewer #3 – Comment 4:\n“- In section VIII in the supplementary materials and in the caption of figure S11 the\nauthors refer to \"32 microphones for excitation\" did the author mean sound sources?”\nResponse from authors:\nThank you for highlighting this point. We agree that the original phrasing was\npotentially ambiguous. To clarify, the “32 microphones” referred to here are indeed\nfunctioning as sound sources. Accordingly, we have revised the manuscript to explicitly\nstate: “32 microphones as sound sources to excite the field” and “32 microphones are\nused as sound sources to generate excitation signals.”\nReviewer #3 – Comment 5:\n“- The authors should also carefully read the manuscript for typos and grammar:\n(for example) \"cease\" instead of \"crease\" in figure 2 caption, \"negligible\" instead of\n\"neglectable\", and \"the phase of the acoustic\" instead of \"the phase of he acoustic”\nResponse from authors:\nWe thank you for carefully reading the manuscript and for pointing out these specific\ntypographical and grammatical errors, as well as for the suggestion to proofread the\nentire text. We have performed a thorough review of the full manuscript and corrected\nthe errors mentioned (e.g., \"crease\" to \"cease\", \"neglectable\" to \"negligible\"), along\nwith any other minor issues we identified (e.g. \"degeneracies\" to \"degeneracy\"). We\nappreciate this helpful feedback, which has enhanced the clarity and quality of our\nmanuscript.\nReviewer #3 – Comment 6:\n“In conclusion, I think this paper is significant and could be of interest to the general\nreadership of Nature Communications after minor revisions.”\nResponse from authors:\nWe sincerely thank you for your time and valuable insights, and for the positive\nassessment that our work is significant and suitable for the readership of Nature\nCommunications. We are very grateful for the constructive suggestions provided\nthroughout the review, which have been instrumental in improving the manuscript. We\nhave carefully addressed all the points raised and believe the revisions have\nstrengthened the paper. Thank you once again for your support and guidance."}]

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