Magnetic fluctuations in Pb$_9$Cu(PO$_4$)$_6$O

The hope that copper doped lead apatite Pb$_9$Cu(PO$_4$)$_6$O is a room-temperature superconductor has largely been dashed by global research efforts. Nevertheless, according to the current state of knowledge, the material has interesting magnetic properties, and research groups around the world have prepared high quality samples. We use a fluctuation exchange approximation (FLEX) approach to study the magnetic tendencies in Pb$_9$Cu(PO$_4$)$_6$O. We find ferromagnetic fluctuations very close to the filling of the stoichiometric compound which can be understood from Fermi surface nesting at the M point. This is similar to the one-band triangular lattice Hamiltonian at three-quarter filling. Interestingly, the special kz dependence of the Pb$_9$Cu(PO$_4$)$_6$O band structure makes it very sensitive to doping. Only slight charge doping switches between antiferromagnetic and ferromagnetic fluctuations. If the material could become superconducting, it might be easily switchable between singlet and triplet superconductivity.