  We investigate magnetic, charge and transport properties of hexagonal
graphene nanoflakes (GNFs) connected to two metallic leads by using the
functional renormalization group (fRG) method. The interplay between the
on-site and long-range interactions leads to a competition of semimetal (SM),
spin density wave (SDW), and charge-density-wave (CDW) phases. The ground-state
phase diagrams are presented for the GNF systems with screened realistic
long-range electron interaction [T. O. Wehling, et. al., Phys. Rev. Lett. 106,
236805 (2011)], as well as uniformly screened long-range Coulomb potential
$\propto 1/r$. We demonstrate that the realistic screening of Coulomb
interaction by $\sigma$ bands causes moderate (strong) enhancement of critical
long-range interaction strength, needed for the SDW (CDW) instability, compared
to the results for the uniformly screened Coulomb potential. This enhancement
gives rise to a wide region of stability of the SM phase for realistic
interaction, such that freely suspended GNFs are far from both SM-SDW and
SM-CDW phase-transition boundaries and correspond to the SM phase. Close
relation between the linear conductance and the magnetic or charge states of
the systems is discussed. A comparison of the results with those of other
studies on GNFs systems and infinite graphene sheet is presented.
