The hope that copper-doped lead apatite ${\mathrm{Pb}}_9\mathrm{Cu}{\left({\mathrm{PO}}_4\right)}_6\mathrm{O}$ is a room-temperature superconductor has largely been dashed by global research efforts. Nevertheless, the material has interesting magnetic properties, and research groups around the world have prepared high-quality samples. We use a fluctuation exchange (FLEX) approximation approach to study the magnetic tendencies in ${\mathrm{Pb}}_9\mathrm{Cu}{\left({\mathrm{PO}}_4\right)}_6\mathrm{O}$. We find ferromagnetic (FM) fluctuations very close to the filling of the stoichiometric compound which can be understood from Fermi surface nesting at the $M$ point. This is like the one-band triangular lattice Hamiltonian at $\frac{3}{4}$ filling. Interestingly, the special $k_z$ dependence of the ${\mathrm{Pb}}_9\mathrm{Cu}{\left({\mathrm{PO}}_4\right)}_6\mathrm{O}$ 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.

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• Electrical properties
• First-principles calculations
• Magnetic susceptibility
• Van Hove singularity
• Electronic structure
• High-temperature superconductors
• Superconductors
• Triangular lattice
• Density functional theory
• Hubbard model
• Magnetism
• Tight-binding model