Level Repulsion and Band Sorting in Phononic Crystals

摘要

In this paper we consider the problem of avoided crossings (level repulsion)in phononic crystals and suggest a computationally efficient strategy todistinguish them from normal cross points. This process is essential for thecorrect sorting of the phononic bands and, subsequently, for the accuratedetermination of mode continuation, group velocities, and emergent propertieswhich depend on them such as thermal conductivity. Through explicit phononiccalculations using generalized Rayleigh quotient, we identify exact locationsof exceptional points in the complex wavenumber domain which results in levelrepulsion in the real domain. We show that in the vicinity of the exceptionalpoint the relevant phononic eigenvalue surfaces resemble the surfaces of a 2 by2 parameter-dependent matrix. Along a closed loop encircling the exceptionalpoint we show that the phononic eigenvalues are exchanged, just as they are forthe 2 by 2 matrix case. However, the behavior of the associated eigenvectors isshown to be more complex in the phononic case. Along a closed loop around anexceptional point, we show that the eigenvectors can flip signs multiple timesunlike a 2 by 2 matrix where the flip of sign occurs only once. Finally, weexploit these eigenvector sign flips around exceptional points to propose asimple and efficient method of distinguishing them from normal crosses and ofcorrectly sorting the band-structure. Our proposed method is roughly anorder-of magnitude faster than the zoom-in method and correctly identifies >97% of the cases considered. Both its speed and accuracy can be furtherimproved and we suggest some ways of achieving this. Our method is general and,as such, would be directly applicable to other eigenvalue problems where theeigenspectrum needs to be correctly sorted.