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

Abstract

We investigate various types of quantum entanglement in the octanuclear heterometallic $3d/4f$ complexes denoted as Ni${}_4^{2+}$Gd${}_4^{3+}$ 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 $\{{\text{Ni}}_2^{2+}{\text{Gd}}_2^{3+}\}$ 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$-$3 tangle, and 2$-$2 tangle entanglements in two families of Ni${}_4^{2+}$Gd${}_4^{3+}$ complexes: $(1)$ without anisotropy and $(2)$ with anisotropy. Complex $(1)$ 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 $(2)$ 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${}_4^{2+}$Gd${}_4^{3+}$ 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.

13 pages, 7 figures

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