  We theoretically study the competition between two possible exotic
superconducting orders that may occur in graphene-like systems, assuming
dominant nearest-neighbor attraction: the gapless hidden superconducting order,
which renormalizes the Fermi velocity, and the Kekule order, which opens a
superconducting gap. We perform an analysis within the mean-field theory for
Dirac electrons, at finite-temperature and finite chemical potential, as well
as at half filling and zero-temperature, first excluding the possibility of the
coexistence of the two orders. In that case, we find the dependence of the
critical (more precisely, crossover) temperature and the critical interaction
on the chemical potential. As a result of this analysis, we find that the
Kekule order is preferred over the hidden order at both finite temperature and
finite chemical potential. However, when the coexistence of the two
superconducting orders is allowed, using the coupled mean-field gap equations,
we find that above a critical value of the attractive interaction a mixed phase
sets in, in which these orders coexist. We show that the critical value of the
interaction for this transition is greater than the critical coupling for the
hidden superconducting state in the absence of the Kekule order, implying that
there is a region in the phase diagram where the Kekule order is favored as a
result of the competition with the hidden superconducting order. The latter,
however, eventually sets in and coexists with the Kekule state. According to
our mean-field calculations, the transition from the Kekule to the mixed phase
is of the second order, but it may become first order when fluctuations are
considered. Finally, we investigate whether these phases could be possible in
honeycomb superlattices of self-assembled semiconducting nanocrystals, which
have been recently experimentally realized with CdSe and PbSe.
