The question of avoided crossing arises frequently in the study of condensed matter systems and always seems surprising to younger graduate students. In condensed matter physics, avoided crossings are ubiquitous, arising in phonon spectra, electronic band spectra and often when different types of quasi-particles interact.
Let me first say that avoided crossings are the norm and degenerate energies only occur because of some strict constraint (usually due to symmetry). As an example, let’s take the simple s-d Hamiltonian of a solid:
where the creates an electron in an s-orbital, creates an electron in a d-orbital, is the hopping parameter, is the energy of an electron in a d-orbital and is the hopping amplitude from an s- to a d-orbital. One can obtain the energy dispersion of this Hamiltonian by diagonalizing the following matrix:
Here are the resulting bands for a couple different values of (click to enlarge):
While in the literature this is usually called a gap, this is nothing more than an avoided crossing and will occur for any finite . Again, this avoided crossing is not limited to electron energy bands, but can occur in many situations. Here is an old experiment by Mooradian and Wright from the 60’s which shows coupling between a plasmon and an optical phonon (click to enlarge):
One can very clearly see the plasmon (the broad peak) “becomes a logitudinal optical phonon” (the narrow peak) and the optical phonon “becomes the plasmon”. Notably, one can see that there is no coupling to the other (transverse) optical phonon because of symmetry reasons (plasmons cannot couple to transverse phonons), and it therefore stays put.