Strongly correlated Dirac electrons and f-wave superconductivity in Ga- herbertsmithite

Herbertsmithite ZnCu$_3$(OH)$_6$C$_{12}$ is essentially the only real-world realization of the ideal single-orbital Kagome model. Being half-doping, it is a Mott insulator. In the nn $p-d$ TB model, it maps exactly onto a single $s$-orbital Kagome Hamiltonian, in particular, exhibits topologically protected Dirac points (DP) at the 4/3 doping. We propose to achieve this doping by substituting Ga for Zn. Such Ga-herbertsmithite (GHS) would be a rare example of a material with strongly correlated Dirac electrons at symmetry-protected locations in the Brillouin zone. We have investigated GHS by means of DFT, TB-DCA and the Slave Bosons approaches and searched for Mott and/or charge order instabilities, and found that it remains metallic and uniform, retaining the DPs. Such a metal with strongly correlated DP electrons would have rather unique topological, magnetic and transport properties. In particular, we show analytically and using fRG that when back-doped with Zn, GHS would harbor unconventional spin-fluctuation driven superconductivity which by symmetry must be $f$-wave of the $+-+-+-$ type.