Theory of non-Hermitian topological whispering gallery

The research explores the non-Hermitian topological aspects of acoustic systems, inspired by recent experimental work using thermoacoustic coupling in sonic lattices coated with electrically biased carbon nanotube films. The authors aim to understand the complex interplay between geometry and acoustic gain in such systems. The lack of acoustic analogs to lasers (sasers) motivates the search for artificial acoustic gain media, crucial for advancing sound-based technologies such as imaging and communication. The study focuses on valley-Hall effects, a frontier in condensed matter physics, characterized by valley-Hall topological insulators capable of confining states due to their topology. Valley transport is robust under symmetry-preserving imperfections, with topological charges localized around valleys in the Brillouin zone. The latest frontier expands on non-Hermitian topological phases by adding gain or loss channels, leading to complex topology and phenomena challenging conventional bulk-edge correspondence, such as the non-Hermitian skin effect. This work investigates a valley-Hall inspired acoustic system, numerically simulating a sonic lattice with thermoplastic rods coated by carbon nanotube films, which when activated, generate non-Hermitian acoustic contributions. The study uses plane wave expansions and finite element simulations, reconstructing the complex acoustic signatures of a triangular topological whispering gallery sonic crystal, focusing on mode-split resonances and acoustic gain saturation.

The paper reviews existing literature on valley-Hall effects in condensed matter physics and their analogs in classical wave systems (photonics and phononics). It highlights the importance of valley-contrasting physics and the recent advancements in non-Hermitian topology, including the non-Hermitian skin effect and its implications for bulk-edge correspondence. The experimental work by Hu et al. (Nature 597, 655 (2021)) on thermoacoustic topological insulators serves as a key inspiration for the current study. The authors cite numerous papers demonstrating theoretical and experimental work on valley-Hall effects in graphene, photonic crystals, and acoustic systems, as well as studies on non-Hermitian topological phases and the non-Hermitian skin effect.

The study employs three primary numerical methods: Plane Wave Expansion (PWE), finite element simulations (COMSOL), and a k⋅p method. The PWE method is used to calculate the complex band diagrams of the non-Hermitian sonic kagome lattice. The lattice is modeled as a fluid-acoustic periodic system, with the ABS rods treated as rigid bodies and the CNT films as complex fluid layers with complex mass density. The inhomogeneous acoustic wave equation is rewritten as a generalized eigenvalue problem to compute the complex band structure. The k⋅p method provides a semi-empirical approach to calculating the electronic band structure, particularly around the K-point. Finite element simulations (COMSOL) are used for validation and comparison with PWE results, focusing on both bulk and edge states. For larger supercells, required for computing valley-polarized edge dispersion, the PWE method is extended. A rectangular supercell is designed to create a valley-Hall zig-zag interface, and the resulting edge dispersion is computed and compared with COMSOL simulations. A Multiple Scattering Theory (MST) approach is employed to analyze the scattering characteristics of finite structures. The active rods are modeled using uniformly distributed acoustic point sources (source-rings) surrounding each cylinder, emulating the activated CNT film coatings. The total pressure field is calculated by summing the incident field and the scattered fields from all cylinders. The MST is used to compute the chiral pressure amplitude spectra for different numbers of activated rows in the WG structure, allowing for the analysis of gain saturation. The k⋅p method is used to derive the Hamiltonian around the K valley, considering the effects of gain and symmetry. The parameters used in the simulations, including material properties and geometric dimensions, are explicitly stated.

The study reveals several key findings: First, the PWE, COMSOL, and k⋅p methods demonstrate good agreement in predicting the band structure and edge states of the non-Hermitian sonic lattice, validating the model. Second, the numerical simulations successfully reproduce the experimentally observed valley-polarized edge states within the topological band gap, highlighting the robustness of the topological protection. Third, the MST simulations accurately predict the mode-splitting of whispering gallery resonances in the non-Hermitian structure, exhibiting good agreement with both experimental and finite element data. Fourth, the study reveals a gain saturation effect in the mode-split amplitudes. The amplitude increases with the number of activated rows until a threshold (four active rows in this case) is reached, beyond which adding more gain does not enhance the amplitude of the WG edge states, highlighting a limitation inherent to the system's geometry. The simulations show that the clockwise and counterclockwise WG modes are carried by the respective valley-polarized edge states. The pressure field distributions and phase profiles are computed using the MST, demonstrating chiral symmetry breaking due to the engineered gain phase texture. The k⋅p analysis indicates that the system remains quasi-Hermitian at moderate gain levels, allowing the application of bulk-edge correspondence. The authors' model successfully connects the valley-Hall effect and non-Hermitian WG physics, demonstrating how adding gain produces amplifying topological edge states that carry the chiral WG modes. The precise relation between these phenomena is discussed in detail.

The findings demonstrate the successful numerical modeling of a non-Hermitian topological whispering gallery resonator, confirming experimental observations and providing a deeper theoretical understanding. The observed gain saturation effect is a significant result, suggesting a limit to enhancing the system's performance solely by increasing gain. This limitation is attributed to the finite penetration depth of the surface states into the bulk of the lattice. The agreement between the different numerical methods validates the approach and confirms the robustness of the topological protection. The detailed investigation of the chiral symmetry breaking using the MST offers valuable insights into the underlying mechanisms. The study bridges the gap between the theoretical understanding of valley-Hall effects and the experimental realization of non-Hermitian topological phenomena in acoustics, opening avenues for further research into acoustic wave manipulation and sensing technologies.

This work provides a comprehensive theoretical and numerical investigation of non-Hermitian topological whispering gallery modes in an acoustic system. The authors successfully modeled the system using several numerical methods, accurately predicting key experimental features such as mode splitting and gain saturation. The findings underscore the potential of non-Hermitian topological systems for applications in acoustic devices and sensing. Future work could explore different geometries, gain mechanisms, and materials to optimize the performance and expand the functionalities of such devices.

The study primarily relies on numerical simulations. While the results show good agreement with experimental data, further experimental validation would strengthen the conclusions. The model simplifies the system by treating the ABS rods as perfectly rigid and the CNT films as homogeneous complex fluid layers. More realistic material models could improve the accuracy of the simulations. The gain saturation effect is investigated within a specific range of parameters, and exploring a broader parameter space could reveal more complex behavior. Finally, the k⋅p analysis is a perturbation method, therefore it is only accurate around the K point and moderate gain levels.