The square-kagome lattice Heisenberg antiferromagnet is a highly frustrated Hamiltonian whose material realizations have been scarce. We theoretically investigate the recently synthesized ${\mathrm{Na}}_6{\mathrm{Cu}}_7{\mathrm{BiO}}_4{\left(\mathrm{PO}\right)}_4{)}_4{\mathrm{Cl}}_3$ where a ${\mathrm{Cu}}^{2+}$ spin-$1/2$ square-kagome lattice (with a six site unit cell) is decorated by a seventh magnetic site alternatingly above and below the layers. The material does not show any sign of long-range magnetic order down to 50 mK despite a Curie-Weiss temperature of $-212\mathrm{K}$ indicating a quantum paramagnetic phase. Our DFT energy mapping elicits a purely antiferromagnetic Hamiltonian that features longer range exchange interactions beyond the pure square-kagome model and, importantly, we find the seventh site to be strongly coupled to the plane. We combine two variational Monte Carlo approaches, pseudofermion/Majorana functional renormalization group and Schwinger-Boson mean field calculations to show that the complex Hamiltonian of ${\mathrm{Na}}_6{\mathrm{Cu}}_7{\mathrm{BiO}}_4{\left(\mathrm{PO}\right)}_4{)}_4{\mathrm{Cl}}_3$ still features a nonmagnetic ground state. We explain how the seventh ${\mathrm{Cu}}^{2+}$ site actually aids the stabilization of the disordered state. We predict static and dynamic spin structure factors to guide future neutron scattering experiments.

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• Exchange interaction
• Frustrated magnetism
• Magnetic susceptibility
• Heisenberg model
• Many-body techniques
• Materials modeling