Quantum entanglement generation in heterometallic Ni 4 2 + Gd 4 3 + complexes

Abstract

We investigate various types of quantum entanglement in the octanuclear heterometallic $3d/4f$ complexes denoted as ${\mathrm{Ni}}_{4}^{2+}{\mathrm{Gd}}_{4}^{3+}$ 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 {${\mathrm{Ni}}_{2}^{2+}{\mathrm{Gd}}_{2}^{3+}$} 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 ${\mathrm{Ni}}_{4}^{2+}{\mathrm{Gd}}_{4}^{3+}$ complexes: (1) without anisotropy and (2) with anisotropy. Complex (1) exhibits strong bipartite entanglement between Ni ions, which persists up to $T\ensuremath{\approx}3.0\phantom{\rule{0.28em}{0ex}}\text{K}$ and $B\ensuremath{\approx}4.0\phantom{\rule{0.28em}{0ex}}\text{T}$, but shows significantly weaker tetrapartite entanglement and vanishing bipartite entanglement between $\mathrm{Gd}\ensuremath{\cdots}\mathrm{Gd}$ and $\mathrm{Ni}\ensuremath{\cdots}\mathrm{Gd}$ 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 ${\mathrm{Ni}}_{4}^{2+}{\mathrm{Gd}}_{4}^{3+}$ 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.