Genuine tripartite entanglement in a mixed spin-(1/2,1) Heisenberg tetramer

Abstract: A genuine tripartite entanglement of a mixed spin-(1/2,1) Heisenberg tetramer is rigorously analyzed in a presence of external magnetic field. The couple of mixed spin-(1/2,1) dimers is arranged in a perfect rectangular square plaquette involving two nonequivalent Heisenberg exchange couplings $J$ and $J_1$. The degree of a genuine tripartite entanglement is evaluated according to the genuine tripartite negativity ${\cal N}{ABC}$ defined as a geometric mean of all possible bipartite negativities corresponding to a decomposition into a single spin and the remaining spin dimer ${\cal N}{A|BC}$, ${\cal N}{B|AC}$ and ${\cal N}{C|AB}$ after degrees of freedom of the last fourth spin $D$ are traced out. Due to the symmetry of a mixed spin-(1/2,1) Heisenberg tetramer two different genuine tripartite negativities for the trimeric system $1/2!-!1!-!1$ and $1/2!-!1/2!-!1$ were identified. It was found that the genuine tripartite negativity for the interaction ratio $J_1/J!<!1$ becomes nonzero solely in the tripartite system $1/2!-!1!-!1$ at low-enough magnetic fields. The opposite interaction limit $J_1/J!>!1$ gives rise to the nonzero genuine tripartite negativity in both tripartite systems in a presence of external magnetic field until the classical ferromagnetic state is achieved. It was shown, that the genuine tripartite negativity of a mixed spin-(1/2,1) Heisenberg tetramer can be detected also at nonzero temperatures. An enhancement of the thermal genuine tripartite negativity through the enlargement of the total spin number of a tripartite system is evidenced. The correlation between the bipartite negativity of two spins and the genuine tripartite negativity is discussed in detail.