Robust global tripartite entanglement in a mixed spin-($1$,$1/2$,$1$) Heisenberg trimer

We rigorously analyze the global tripartite entanglement in a Heisenberg trimer with mixed spins-($1$,$1/2$,$1$) under varying exchange couplings between dissimilar and identical spins, magnetic fields, and temperatures. The global tripartite entanglement is quantified using the geometric mean of all three bipartite contributions, evaluated through negativity. We precisely map the regions of parameter space in which the trimer system exhibits spontaneous global entanglement. In addition, we classify the nature of the tripartite entangled states based on the distribution of reduced bipartite negativities among the spin dimers in the trimer. We further examine the thermal stability of global tripartite entanglement throughout the full parameter space. Special attention is given to the theoretical prediction of thermal robustness in a real three-spin complex, [Ni(bapa)(H$_2$0)]$_2$Cu(pba)(ClO$_4$)$_2$, where bapa stands for bis($3$-aminopropyl)amine, and pba denotes $1$,$3$-propylenebis(oxamato), which serves as an experimental realization of the mixed-spin ($1$,$1/2$,$1$) Heisenberg trimer. Notably, global entanglement in this system is predicted to persist up to approximately $100$ K and magnetic fields approaching $210$ T. Moreover, we uncover a thermally induced activation of robust global entanglement in regions where the ground state is biseparable. The magnitude of this thermal entanglement is remarkably high, nearly reaching a value of $1/2$, which has not been reported before. Finally, we propose a connection between the theoretically predicted tripartite entanglement, quantified via negativity derived from the density matrix, and the quantities measured directly or indirectly from various experiments.