Landscape of incompressible crystals of hard-core bosons on the square-kagome lattice

We investigate a hard-core boson model on the square-kagome lattice using hierarchical mean-field theory beyond the conventional unit-cell description. We show that the conventional description of the square-kagome lattice based on its smallest unit cell does not fully capture the hierarchy of incompressible states supported by the lattice, but only captures the two compact-localizedstate-based phases at densities 5/6 and 2/3. Within our approach, we reproduce these previously established phases and uncover additional incompressible states enabled by enlarging the variational cluster. Among these, the ρ = 3/4 phase is found to be particularly robust, which, through the Matsubara-Matsuda mapping, corresponds to a half-magnetization plateau in the spin-1/2 XXZ model. We further apply our approach to two experimentally relevant square-kagome compounds using exchange parameters obtained from first-principles calculations. The calculated magnetization processes are in good agreement with available experimental results and predict additional plateau structures in these materials.