Dzyaloshinskii-Moriya-driven instabilities in square-kagome quantum antiferromagnets

Decorated square-kagome quantum antiferromagnets provide a natural setting in which strong frustration, lattice decoration, and spin-orbit-induced anisotropy compete on comparable energy scales. Here we show that in Na$_6$Cu$_7$BiO$_4$(PO$_4$)$_4$Cl$_3$ the coupling ($J_{10}$) which links the decorating Cu(3) sites to the square-kagome backbone, stabilizes the gapped quantum-paramagnetic regime, while symmetry-allowed Dzyaloshinskii-Moriya (DM) interactions systematically suppress the minimum spinon gap $\Delta_{\mathrm{spinon}}$ and drive the system toward magnetic condensation. To establish this, we combine ab initio calculation of the DM vectors with a generalized Schwinger-boson self-consistent mean-field theory that treats singlet and triplet hopping/pairing channels on equal footing. As a benchmark, the isotropic square-kagome Heisenberg model exhibits four competing low-energy saddle points distinguished by their Wilson-loop fluxes and by characteristic static and dynamical structure-factor fingerprints. A minimal DM perturbation does not qualitatively reshape this competing landscape, but already enhances the tendency towards order. For the realistic decorated Hamiltonian, finite-size scaling of $\Delta_{\mathrm{spinon}}$ together with momentum-resolved structure factors identifies $J_{10}$ (exchange with decorating Cu) as the control parameter of the gapped regime and shows that the full symmetry-allowed DM pattern shifts the system further toward condensation. Our results place Na$_6$Cu$_7$BiO$_4$(PO$_4$)$_4$Cl$_3$ in close proximity to a magnetic instability and provide experimentally testable predictions for anisotropy-enhanced soft modes in decorated square-kagome materials.