Theory of non-Hermitian topological whispering gallery

The paper investigates non-Hermitian, valley-Hall-inspired topological acoustics as an analogue to optical amplification and lasers but for sound at kHz–MHz frequencies, where no direct acoustic gain medium exists. It places the work within the context of valley-Hall topological insulators that support robust, valley-polarized edge transport without breaking time-reversal symmetry, and surveys developments in non-Hermitian topology (including the skin effect) that challenge conventional bulk–edge correspondence. Motivated by Hu et al. (Nature 2021), the authors model thermoacoustic gain via carbon nanotube (CNT) coatings on ABS rods in a kagome (triangular) sonic lattice. They aim to clarify how geometry and gain produce complex bulk and edge spectra, whispering-gallery (WG) mode splitting with controlled chirality, and whether increasing the number of activated (gain) rows leads to amplitude saturation at the topological interface. The study’s purpose is to provide a predictive theoretical and numerical framework to guide future explorations of thermoacoustic topological effects for acoustic communication and sensing.

The introduction reviews valley-Hall physics in condensed matter and its classical analogues, noting valley Chern numbers localized near Brillouin-zone corners and their use for one-way transport, beam splitting, and refraction control. It surveys non-Hermitian topological phases that expand conventional classifications and can disrupt standard bulk–edge correspondence, highlighting the non-Hermitian skin effect and related theoretical developments in various platforms. The work is directly inspired by Hu et al. (Nature 2021), who experimentally realized non-Hermitian topological WG acoustics using CNT-coated rods driven by electric currents to provide thermoacoustic gain.

The study is theoretical and numerical, combining: (1) Plane Wave Expansion (PWE) to compute complex bulk and projected edge band structures for a non-Hermitian kagome (triangular) sonic lattice of rigid ABS rods coated by CNT films modeled as complex fluid layers. The acoustic medium obeys a linear inhomogeneous wave equation with spatially periodic bulk modulus and mass density; the CNT coating is represented by a complex mass density ρ_CNT = ρ0(1 + iβ), with β controlling gain (β = 0.05 used). PWE casts the problem into a generalized eigenvalue problem in reciprocal space using Fourier expansions of material parameters and Bloch pressure fields. A supercell PWE is built to model a zig-zag valley-Hall interface between two lattices of opposite valley phases to obtain edge dispersions. (2) k·p method around the K valley to derive an effective Dirac Hamiltonian capturing the gapped Dirac cones when inversion symmetry is broken (via trimer rotation) and incorporating small non-Hermitian (gain) terms. Symmetry (C3v) constraints simplify the Hamiltonian; parameters (Dirac velocity and small complex shifts) are fitted to finite-element data to match dispersion near K. (3) Multiple Scattering Theory (MST) for finite structures to model WG resonators with many trimers. Rigid cylinders scatter incident fields; total fields are expanded in Bessel/Hankel series with coefficients determined by transfer-matrix relations for rigid cylinders. To emulate active CNT coatings, each rod is surrounded by a ring of np monopole point sources (np = 18), with phase patterns per rod within a trimer set to φ1 = 0, φ2 = 2π/3, φ3 = −2π/3 (full gain cycle φ = 2π) to enforce chiral phase control. The triangular WG cavity consists of two sonic crystals of opposite valley phase forming a closed valley interface; rows of trimers on both sides of the interface are activated to supply gain. Pressure amplitudes are computed along equidistant points at the gallery edge to extract WG spectra and mode splitting. Parameters: air ρ0 = 1.22 kg/m^3, c0 = 342 m/s; cylinder radius R = 0.30 cm; trimer side D = 0.69 cm; lattice constant a = 2.17 cm; non-Hermiticity β = 0.05. Finite-element simulations (COMSOL) validate PWE and MST predictions; experimental spectra from Hu et al. are reused for comparison.

• Non-Hermitian kagome sonic lattice (β = 0.05) exhibits a gapped Dirac dispersion at K with negative imaginary parts of eigenfrequencies, indicating amplification; PWE, k·p (near K), and COMSOL agree on real-part gaps and complex bands. - Valley-projected edge states appear inside the bulk gap for a zig-zag interface between opposite valley phases; supercell PWE and COMSOL agree on two counterpropagating edge modes (positive/negative-type interfaces). - Chiral whispering-gallery (WG) modes in a triangular cavity are realized by imposing a full gain-cycle phase texture (φ = 2π) across source rings in each trimer, breaking chiral symmetry of the m = 27 WG mode and yielding clear mode splitting into clockwise and counterclockwise chiral modes tied to K/K′ valley polarization. - MST, COMSOL, and experimental data show consistent split peaks in the WG amplitude spectrum; averaging along gallery edges captures both modes with good agreement. - Gain saturation: Increasing the number of active rows of CNT-coated trimers on each side of the interface increases amplitude initially, but the average of the two chiral mode amplitudes saturates once four rows per side are activated (eight rows total), indicating that additional gain beyond the penetration depth of the edge states does not enhance WG amplitude. - Effective k·p model (quasi-Hermitian regime) yields example complex eigenfrequencies near K of f+ ≈ 9316.2 − 15i Hz and f− ≈ 8265.7 − 27.42i Hz, consistent with moderate gain where bulk–edge correspondence remains valid and no skin effect is expected.

The findings link valley-Hall topology with non-Hermitian acoustics: adding gain to valley-Hall sonic lattices amplifies topological edge states that carry WG modes. With an engineered gain-phase texture at the unit-cell level, WG modes become chiral and split according to valley polarization (e.g., clockwise mode from the K valley). In the moderate-gain, quasi-Hermitian regime (β ≈ 0.05), topological characterization via a valley Chern number remains meaningful, bulk–edge correspondence holds, and non-Hermitian skin effects are not expected. The gain-saturation result shows that amplification is limited by the edge-state penetration depth: beyond four activated rows per side in this geometry, additional gain does not reach the interface effectively. These insights guide the design of non-Hermitian topological WG devices for robust acoustic routing, communication, and sensing, and clarify the interplay between geometry, gain distribution, and topological edge transport.

The work provides a theoretical and numerical framework for non-Hermitian topological whispering-gallery acoustics. Using PWE and k·p theory for complex bulk/edge bands and MST with source-ring models for finite structures, the study reproduces and explains chiral WG mode splitting observed experimentally in CNT-coated kagome sonic lattices. A key prediction is gain saturation of WG amplitudes when more than four active rows per side are used, set by the edge-state penetration depth. The approach offers design rules for tailoring mode splitting and saturation thresholds via geometric scaling and gain distribution, paving the way for topological acoustic devices with controllable amplification and chirality. Future work could explore different gain profiles, higher gain regimes beyond quasi-Hermitian limits, alternative lattice symmetries, and integration with sensing/communication platforms.

The study is primarily theoretical/numerical and reuses experimental spectra for comparison rather than reporting new experiments. Modeling assumes rigid ABS cylinders and represents CNT films as either thin complex-fluid layers (PWE) or source-ring monopoles (MST), which approximate thermoacoustic gain. Conclusions rely on a moderate-gain, quasi-Hermitian regime (β ≈ 0.05) where bulk–edge correspondence holds and skin effects are absent; behavior at higher gains is not addressed. Gain is introduced only via coated elements and with a specific phase texture; other gain/loss distributions are not explored. The gain-saturation threshold (four active rows per side) pertains to the specific geometry and parameters studied and may vary with design.