Incommensurate Spin-Density Waves in a Frustrated Maple-Leaf Lattice Ferromagnet

We study how ferromagnetism breaks down in the spin-$\tfrac12$ nearest-neighbor Heisenberg model on the maple-leaf lattice with ferromagnetic $J_t,J_d$ and antiferromagnetic $J_h$, motivated by the mixed ferro-antiferromagnetic interactions in Na$_2$Mn$_3$O$_7$. Exact diagonalization shows that the ferromagnetic boundary does not feature a zero-field spin-nematic phase on the clusters studied here, but an extended regime of incommensurate spin-density-wave correlations with continuously evolving ordering vector. The phase diagram also contains collinear N\'eel, canted $120^\circ$, and hexagonal-singlet regimes, separated by regions that remain difficult to classify from exact diagonalization alone. Variational tests of fully symmetric Gutzwiller-projected Abrikosov-fermion U(1) and $\mathbb{Z}_2$ states find no competitive spin-liquid description of the interior unresolved regions. By contrast, on the ruby-lattice boundary we identify a point between the collinear N\'eel and hexagonal-singlet phases where a projected $\mathbb{Z}_2$ Ansatz reproduces the finite-size energy and spin correlations with good accuracy.