  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.
