  We theoretically study the excitation spectrum of a two-dimensional Dirac
semimetal in the presence of an incommensurate potential. Such models have been
shown to possess magic-angle critical points in the single particle
wavefunctions, signalled by a momentum space delocalization of plane wave
eigenstates and flat bands due to a vanishing Dirac cone velocity. Using the
kernel polynomial method, we compute the single particle Green's function to
extract the nature of the single particle excitation energy, damping rate, and
quasiparticle residue. As a result, we are able to clearly demonstrate the
redistribution of spectral weight due to quasiperiodicity-induced downfolding
of the Brillouin zone creating minibands with effective mini Brillouin zones
that correspond to emergent superlattices. By computing the damping rate we
show that the vanishing of the velocity and generation of finite density of
states at the magic-angle transition coincides with the development of an
imaginary part in the self energy and a suppression of the quasiparticle
residue that vanishes in a power law like fashion. Observing these effects with
ultracold atoms using momentum resolved radiofrequency spectroscopy is
discussed.
