  Dirac fermions exist in many solid state systems including graphene, silicene
and other two dimensional membranes such as are found in group VI
dichalcogenides, as well as on the surface of some insulators where such states
are protected by topology. Coupling of those fermions to phonons introduces new
structures in their dispersion curves and, in the case of massive Dirac
fermions, can shift and modify the gap. We show how these changes present in
angular-resolved photoemission spectroscopy of the dressed charge carrier
dispersion curves and scanning tunneling microscopy measurements of their
density of states. In particular we focus on the region around the band gap. In
this region the charge carrier spectral density no longer consists of a
dominant quasiparticle peak and a smaller incoherent phonon related background.
The quasiparticle picture has broken down and this leads to important
modification in both dispersion curves and density of states.
