  Despite being relevant to better understand the properties of honeycomb-like
systems, as graphene-based compounds, the electron-phonon interaction is
commonly disregarded in theoretical approaches. That is, the effects of phonon
fields on \textit{interacting} Dirac electrons is an open issue, in particular
when investigating long-range ordering. Thus, here we perform unbiased quantum
Monte Carlo simulations to examine the Hubbard-Holstein model (HHM) in the
half-filled honeycomb lattice. By performing careful finite-size scaling
analysis, we identify semimetal-to-insulator quantum critical points, and
determine the behavior of the antiferromagnetic and charge-density wave phase
transitions. We have, therefore, established the ground state phase diagram of
the HHM for intermediate interaction strength, determining its behavior for
different phonon frequencies. Our findings represent a complete description of
the model, and may shed light on the emergence of many-body properties in
honeycomb-like systems.
