Level Repulsion and Band Sorting in Phononic Crystals

Abstract

In this paper we consider the problem of avoided crossings (level repulsion) in phononic crystals and suggest a computationally efficient strategy to distinguish them from normal cross points. This process is essential for the correct sorting of the phononic bands and, subsequently, for the accurate determination of mode continuation, group velocities, and emergent properties which depend on them such as thermal conductivity. Through explicit phononic calculations using generalized Rayleigh quotient, we identify exact locations of exceptional points in the complex wavenumber domain which results in level repulsion in the real domain. We show that in the vicinity of the exceptional point the relevant phononic eigenvalue surfaces resemble the surfaces of a 2 by 2 parameter-dependent matrix. Along a closed loop encircling the exceptional point we show that the phononic eigenvalues are exchanged, just as they are for the 2 by 2 matrix case. However, the behavior of the associated eigenvectors is shown to be more complex in the phononic case. Along a closed loop around an exceptional point, we show that the eigenvectors can flip signs multiple times unlike a 2 by 2 matrix where the flip of sign occurs only once. Finally, we exploit these eigenvector sign flips around exceptional points to propose a simple and efficient method of distinguishing them from normal crosses and of correctly sorting the band-structure. Our proposed method is roughly an order-of magnitude faster than the zoom-in method and correctly identifies > 97% of the cases considered. Both its speed and accuracy can be further improved and we suggest some ways of achieving this. Our method is general and, as such, would be directly applicable to other eigenvalue problems where the eigenspectrum needs to be correctly sorted.

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• Linking scalar elastodynamics and non-Hermitian quantum mechanics
• Symmetry and Degeneracy of Phonon Modes for Periodic Structures with Glide Symmetry
• Exceptional points and scattering of discrete mechanical metamaterials
• Reduced Order Modeling of Dynamic Mechanical Metamaterials for Analysis of Infinite and Finite Systems
• Reduced order derivation of the two-dimensional band structure of a mixed-mode resonator array
• In-Plane Linear Stress Waves in Layered Media: I. Non-Hermitian Degeneracies and Modal Chirality