# Level Repulsion and Band Sorting in Phononic Crystals

## TL;DR

This paper introduces a fast, accurate method for distinguishing level repulsion from normal crossings in phononic crystals, crucial for correct band sorting and property analysis, by analyzing eigenvector sign flips around exceptional points.

## Contribution

The paper proposes a novel, efficient technique to identify exceptional points in phononic band structures, improving speed and accuracy over existing methods.

## Key findings

- The method correctly identifies over 97% of cases.
- Eigenvector sign flips are more complex in phononic crystals than in simple matrices.
- The approach is applicable to general eigenvalue problems.

## 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.