Acoustic Metamaterials with Positive and Negative Couplings: Modular and One Piece Architectures for Topological Models

TL;DR

The paper tackles realizing tight-binding topological models in acoustic metamaterials using tube-coupled H-resonators, enabling both positive and negative couplings. It introduces two fabrication routes—modular reconfigurable resonators and monolithic one-piece printing—and demonstrates how coupling-length tuning ($\text{CL}$) and Total Coupling Area ($\text{TCA}$) control mitigate frequency detuning ($\epsilon$) and preserve particle-hole symmetry. Through experimental and numerical study of SSH and Kitaev chains, including a Kitaev interface, the work shows clear edge and midgap interface states and confirms the robustness of topological features across architectures and scales. The approach provides design principles for accurate acoustic realizations of complex tight-binding Hamiltonians with potential applications in robust acoustic waveguiding and sensing.

Abstract

We describe two 3D-printing approaches for realizing tight-binding models in acoustic metamaterials using H-shaped resonators: a modular system with tunable interconnections and an integrated one-piece design for reducing dissipation. The platform supports both positive and negative coupling through geometric control, enabling accurate acoustic analogs of topological models. By tuning the coupling length (CL), we eliminate detuning effects and preserve particle-hole symmetry. We further quantify the influence of the Total Coupling Area (TCA) on band topology and derive conditions for constant-area coupling. The system was tested on SSH and Kitaev chains, revealing midgap edge and interface states, confirming topological behavior in both configurations.

Figures (6)

• Figure 1: H-resonator design and spectra.(a) Cad model of double (top) and single (bottom) bridge H-resonator with dimensions $h_1 = 40$ mm, $w_1 = 20$ mm, $wt = 4$ mm, $d = 6$ mm, & $w_2 = 10$ mm. (b) To-scale 3D printed resonator comparison. Larger double-bridge resonator (top) printed hollow then sealed with a cap using puddy. Smaller single-bridge resonator (bottom) printed fully encased. (c) Resonance spectrum and corresponding modes for the double-bridge resonator. Pressure legend inlayed in the top right. (d) Resonance spectrum and corresponding modes for the single-bridge resonator. (c) & (d) Selected $f_o$ mode that all later systems are designed around are highlighted in green. For the modular systems $f_o^m = 2.38$ kHz and for the single-shot systems $f_o^s = 3.38$ kHz
• Figure 2: Positive/negative coupling schematic and hybridization diagrams with simulated and experimental dimer results.(a) Internal reference frame of resonators denoted with orange and purple. Positive coupling is defined as coupling orange to orange. Negative coupling is defined as coupling Purple to Orange. (b) Hybridization diagrams for the H-resonator describing the symmetric and antisymmetric modes corresponding to positive (left) and negative (right) couplings corresponding to $H_C^+$ and $H_C^-$ respectively Eq.\ref{['eq:2']}(c) CAD and experimental set up for both double (left) and single (right) bridge dimer apparatus. (d) Resonance spectra for the dimer as a function of CW. Each plot represents a different geometry defined by a unique CL value. Symmetric plot corresponding to $\epsilon = 0$ (left) and asymmetric plot corresponding to $\epsilon \ne 0$ (center & right). (e) Experimental measurements of dimer apparatus demonstrating symmetry at various coupling widths for the modular system with CL = 109 mm. (f) Experimental measurements of dimer apparatus demonstrating symmetry at various coupling widths for single-shot dimer CL=25mm.
• Figure 3: SSH schematic, simulation, and experimental measurements.(a) SSH coupling schematic corresponding to \ref{['eq:ssh']}. (b) Simulated geometry with acoustic pressure profiles corresponding to the edge modes located at the orange and purple stars in (c) and (d). The geometry in the orange (purple) box corresponds to the non-constant (constant) area parameterization of the coupling widths. (c) SSH band spectrum for the non-constant area coupling parameterization. (d) SSH band spectrum for constant area coupling parameterization. Green box corresponds to the experimentally realized parameters as seen in (e). (c,d) Red dots indicate topological modes localized to the edge as seen in (b). (e) Experimental single-shot SSH system comprised of 8 resonators. (f) Band spectrum for experimental system depicted in (e). Edge modes (purple peaks) centered between bulk peaks (red).
• Figure 4: Kitaev chain schematic, simulation design, and results. (a) Acoustic Kitaev chain schematic. Thick and think red lines correspond to alternating couplings strengths $t_1$ & $t_2$. Thick blue line correspond to the onsite hopping potential $\mu$. Unit cell is circled in purple. (b) simulated geometry for the modular method. CW parameters $t_1$, $t_2$, and $\mu$ are labeled. (c) Band spectrum for the Kitaev chain system. Each vertical column of points corresponds to a different $\mu$ value. Topological modes are denoted in red while bulk modes are depicted in black. The mode circled in Green corresponds to the pressure profile depicted in the green box on the right.
• Figure 5: Classical Kitaev chain schematic and simulated band spectrum/topological modes.(a) Single layer Kitaev chain schematic corresponding to real-valued $\Delta$. Coupling strengths $t_1, t_2, \mu_1, \text{and } \mu_2$ are labeled. Positive (negative) couplings correspond to Red (blue) lines. (b) Simulated geometry for single-shot Kitaev interface. Coupling widths corresponding to analogous coupling strengths in (a) are labeled. (c) Band spectrum for simulated modular method with 2 mid gap modes marked in green and red. (d) Edge (interface) corresponding to the green (red) point in (c). (e) Band spectrum for simulated single-shot method with 2 mid gap modes marked in green and red. (f) Edge (interface) mode depicted corresponding to the green (red) point in (e).