Quantum Entanglement Generation in the Heterometallic Ni$^\text{2+}_4$Gd$_4^\text{3+}$ Complexes

We investigate various types of quantum entanglement in the octanuclear heterometallic $3d/4f$ complexes denoted as Ni$^{2+}_4$Gd$^{3+}_4$ under an external magnetic field, using the exact diagonalization approach. These molecular magnets, which can be effectively described by Heisenberg spin models, consist of two identical $\{\text{Ni}^{2+}_2\text{Gd}^{3+}_2\}$ cubane subunits bridged by acetate and hydroxide ligands. Our analysis reveals that their magnetization exhibits intermediate plateaus at low temperatures, indicating distinct ground states characteristic of Ni-containing compounds. Using negativity as a measure of quantum entanglement, we examine the influence of single-ion anisotropy and magnetic field on tetrapartite, bipartite, 1$-$3 tangle, and 2$-$2 tangle entanglements in two families of Ni$^{2+}_4$Gd$^{3+}_4$ complexes: $\boldsymbol{(1)}$ without anisotropy and $\boldsymbol{(2)}$ with anisotropy. Complex $\boldsymbol{(1)}$ exhibits strong bipartite entanglement between Ni ions, which persists up to $T \approx 3.0\,\text{K}$ and $B \approx 4.0\,\text{T}$, but shows significantly weaker tetrapartite entanglement and vanishing bipartite entanglement between Gd$\cdots$Gd and Ni$\cdots$Gd pairs. In contrast, complex $\boldsymbol{(2)}$ displays nonzero and sizable values for all types of entanglement considered. These findings emphasis the crucial role of single-ion anisotropy in generating and shaping the entanglement landscape of heterometallic Ni$^{2+}_4$Gd$^{3+}_4$ complexes. Notably, we find that the 1$-$3 tangle entanglement between a Ni ion and the remaining sites in a cubane unit serves as a reliable indicator of ground-state phase transitions, exhibiting distinct changes across phase boundaries irrespective of the presence of single-ion anisotropy.