  In this paper we study transport properties of electrons on the
two-dimensional honeycomb lattice. We consider a half-filled system in the
vicinity of a symmetry-breaking transition from a semimetallic phase towards an
insulating phase with either charge density or spin density wave order. The
effect of either order is to break the sublattice inversion symmetry which
induces a finite gap for the electronic single-particle excitations.
Phenomenologically, such a scenario is described in the framework of a
Gross-Neveu theory. We analyze two related formulations of the model by means
of (i) a controlled renormalization group calculation and (ii) the large-N
method, both of which in combination with a Boltzmann transport equation. We
determine the quantum-critical conductivity and also discuss crossover behavior
from quantum critical behavior into the insulating and/or the semimetallic
phases. We find that at asymptotically low temperatures the quantum-critical
conductivity is given by a temperature independent universal number. Over a
large temperature window the temperature independent quantum critical
conductivity is masked by a logarithmically temperature dependent contribution
due to the marginally irrelevant long-range Coulomb interaction. We discuss
possible origins of this peculiarity in the two complementary formulations of
the model. Furthermore, we consider possible relations of our findings to
recent experiments, with a special emphasis on the
quantum-critical-to-insulator crossover. We find that our results are in
remarkably good qualitative and quantitative agreement with a recent analysis
of the data sets under the hypothesis of an underlying gap in the
single-particle spectrum.
