  Recently, the most intensely studied objects in the electronic theory of
solids have been strongly correlated systems and graphene. However, the fact
that the Dirac bands in graphene are made up of $sp^{2}$-electrons, which are
subject to neither strong Hubbard repulsion $U$ nor strong Hund's rule coupling
$J$ creates certain limitations in terms of novel, interaction-induced physics
that could be derived from Dirac points. Here we propose
GaCu$_{3}$(OH)$_{6}$Cl$_{2}$ (Ga-substituted herbertsmithite) as a correlated
Dirac-Kagome metal combining Dirac electrons, strong interactions and
frustrated magnetism. Using density functional theory (DFT), we calculate its
crystallographic and electronic properties, and observe that it has
symmetry-protected Dirac points at the Fermi level. Its many-body physics is
excitingly rich, with possible charge, magnetic and superconducting
instabilities. Through a combination of various many-body methods we study
possible symmetry-lowering phase transitions such as Mott-Hubbard, charge or
magnetic ordering, and unconventional superconductivity, which in this compound
assumes an $f$-wave symmetry.
