We investigate frustrated magnetism of a corner-connected square-plaquette Heisenberg model with exchange interactions along the edges $\left(J_1\right)$, along interplaquette links $\left(J_2\right)$, and along square diagonals $\left(J_3\right)$. Using Luttinger-Tisza (LT) minimization of the Fourier interaction matrix $\mathcal{J}\left(\mathbf{q}\right)$ together with large-scale Monte Carlo (MC) simulations, we obtain a classical low-temperature magnetic phase diagram in the normalized plane $(J_1/|J_2|,J_3/|J_2\left|\right)$. The two methods play complementary roles: the LT analysis provides the zero-temperature candidate ordering wave vectors, while the MC simulations elucidate the resulting ordering tendencies under the hard-spin constraint at low but finite temperatures. Three regimes emerge at low temperatures: an antiferromagnetic phase, a ferromagnetic phase, and a degenerate spin state with up-up-down-down (uudd) ordering (DS). For the frustrated degenerate spin state, LT exhibits linelike minima along $q_x=\pm q_y$ in the $hk$ plane, revealing a highly degenerate spin configuration that violates the hard-spin constraint. The MC results uncover a state with quasi-two-dimensional ordering. The DS regime is intrinsically multi-$\mathbf{q}$: the ordered texture assembles itself from symmetry-related modes on the lines, producing a “uudd” spin arrangement comprising distinctive strong antiferromagnetic correlations on diagonals, ferromagnetic correlations on corner links, and highly suppressed correlations on the edges. The field-temperature phase diagram for a representative parameter point in the DS regime, determined by MC simulations, yields field-induced distinct regions of negatively and positively correlated layers. These two regions are separated by a curve corresponding to negligible interlayer correlations. Our $\mathrm{LT}+\mathrm{MC}$ framework delivers a well-controlled classical baseline and operational diagnostics for corner-connected frustrated square-plaquette systems, thereby benchmarking future approaches that incorporate quantum-fluctuation effects.

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• Frustrated magnetism
• Magnetism
• Antiferromagnets
• Monte Carlo methods