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The hope that copper doped lead apatite Pb$_9$Cu(PO$_4$)$_6$O is aroom-temperature superconductor has largely been dashed by global researchefforts. Nevertheless, according to the current state of knowledge, thematerial has interesting magnetic properties, and research groups around theworld have prepared high quality samples. We use a fluctuation exchangeapproximation (FLEX) approach to study the magnetic tendencies inPb$_9$Cu(PO$_4$)$_6$O. We find ferromagnetic fluctuations very close to thefilling of the stoichiometric compound which can be understood from Fermisurface nesting at the M point. This is similar to the one-band triangularlattice Hamiltonian at three-quarter filling. Interestingly, the special kzdependence of the Pb$_9$Cu(PO$_4$)$_6$O band structure makes it very sensitiveto doping. Only slight charge doping switches between antiferromagnetic andferromagnetic fluctuations. If the material could become superconducting, itmight be easily switchable between singlet and triplet superconductivity.

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