Evolution of Antiferromagnetism with Hole Doping in HgBa₂Ca₂Cu₃O₈: <i>A Parameter Free Perspective</i>

Since the discovery of the cuprates 33 years ago, connecting their physical properties to their electronic structure has proven extremely challenging to capture within a uniform theoretical picture. Here, by utilizing the recently constructed SCAN metaGGA, we show how the charge, spin and lattice degrees of freedom of HgBa₂Ca₂Cu₃O_8+δ evolve with oxygen hole doping. Both the layer and doping dependence of our theoretically predicted antiferromagnetic order are in good agreement with NMR observations. In particular, a local maximum in the number of holes in the CuO₂ plane is found in agreement with near optimal δ=0.16 doping for high-T_c superconductivity. Additionally, we find the doped interstitial oxygens play a consequential role at the Fermi level throughout the phase diagram, indicating the importance of inter-layer coupling between the CuO₂ planes and the charge reservoir layer.