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.
翻译:在本文中, 我们考虑在声波晶体中避免交叉( 水平反射) 的问题, 并提出一个计算效率高的战略, 将它们与正常交叉点区分开来。 这个过程对于正确排序声波带和随后准确确定模式延续、 组速度和突发特性至关重要, 这取决于它们, 比如热传导性。 通过使用通用Rayleg 商数的清晰声波计算, 我们确定复杂波数域中特殊点的确切位置, 导致真实域中水平反弹速度。 我们显示, 在异常点附近, 相关的声波天价表面类似于 2+2 参数依赖矩阵的表面 。 在闭路循环中, 我们显示, 声平电子数值的精确度会与2+2矩阵的精确度值是相交替的。 然而, 相关的静脉冲因素的行为在声波数域中表现得更复杂。 在一个特殊点周围, 我们发现, egen值值的表面表面表面表面表面表面表面表面表面表面表面表面和结构中, 显示我们最后会以2 手法 。