Multivalued dispersion equation for plasmon-surface optical phonon coupling in graphene/polar substrate system

Dispersion equations are a common paradigm of collective excitation physics. However, in some systems, dispersion equations contain multivalued functions and their solutions are ambiguous. As an example, we consider graphene on a polar substrate where Dirac plasmons are coupled with surface optical phonons. The dispersion equation for this system contains square-root singularity. Using the initial value problem resolves this uncertainty and gives a unique solution. Particularly, we found that lower plasmon-phonon mode, which in terms of dispersion can have a good quality factor, is almost absent in excitation spectra. The physical reason and experimental evidence of the mode collapse are discussed.