  The attractive Hubbard model on the honeycomb lattice exhibits, at
half-filling, a quantum critical point (QCP) between a semimetal with massless
Dirac fermions and an s-wave superconductor (SC). We study the BCS-BEC
crossover in this model away from half-filling at zero temperature and show
that the appropriately defined crossover line (in the interaction-density
plane) passes through the QCP at half-filling. For a range of densities around
half-filling, the ``underlying Fermi surface'' of the SC, defined as the
momentum space locus of minimum energy quasiparticle excitations, encloses an
area which evolves nonmonotonically with interactions. We also study
fluctuations in the SC and the semimetal, and show the emergence of an undamped
Leggett mode deep in the SC. We consider possible implications for experiments
on ultracold atoms and high temperature SCs.
