  The opening of a gap in single-layer graphene is often ascribed to the
breaking of the equivalence between the two carbon sublattices. We show by
angle-resolved photoemission spectroscopy that Ir- and Na-modified graphene
grown on the Ir(111) surface presents a very large unconventional gap that can
be described in terms of a phenomenological "massless" Dirac model. We discuss
the consequences and differences of this model in comparison of the standard
massive gap model, and we investigate the conditions under which such anomalous
gap can arise from a spontaneous symmetry breaking.
