Abstract A model of a spin-1/2 trimer coupled to a magnetic ion with spins ranging from 1/2 to 5/2 is introduced. We numerically calculate tripartite negativity of spin trimer. Numerical results show that the tripartite negativity has a nonzero plateau regardless of spin of the magnetic ion. We focus on tripartite entangled state corresponding to this plateau. As spin and exchange coupling of the magnetic ion increase, the tripartite entangled state becomes increasingly stable with respect to changes in external magnetic field and temperature. This phenomenon originates from the fact that internal magnetic field formed by the magnetic ion compensates the external magnetic field applied to spin trimer.
Abstract A model of a spin-1/2 trimer coupled to a magnetic ion with spins ranging from 1/2 to 5/2 is introduced. We numerically calculate tripartite negativity of spin trimer. Numerical results show that the tripartite negativity has a nonzero plateau regardless of spin of the magnetic ion. We focus on tripartite entangled state corresponding to this plateau. As spin and exchange coupling of the magnetic ion increase, the tripartite entangled state becomes increasingly stable with respect to changes in external magnetic field and temperature. This phenomenon originates from the fact that internal magnetic field formed by the magnetic ion compensates the external magnetic field applied to spin trimer.
The effect of uniaxial single-ion anisotropy on quantum entanglement is rigorously quantified using negativity in a mixed spin-(1,1/2,1) Heisenberg trimer, accounting for different exchange coupling constants between identical and distinct spins. Bipartite negativities between the single-spin entity and the remaining spin dimer are analyzed alongside the global tripartite negativity (gTN) of the whole trimer under the effect of an external magnetic field and both easy-axis and easy-plane types of single-ion anisotropy. Interestingly, the single-ion anisotropy significantly influences the degree of entanglement by altering the stability regions of energetically preferred ground states and it may also introduce additional phases in the overall ground-state phase diagram. Moreover, even an arbitrarily small single-ion anisotropy is sufficient to generate global tripartite entanglement in a state that remains biseparable in the isotropic limit. Finally, the thermal stability of entanglement is discussed in detail, including the emergence of a peculiar local minimum at finite temperatures. The obtained theoretical results may offer deeper insights into bipartite and tripartite entanglement in trimetallic Ni2+-Cu2+-Ni2+molecular compounds.
We rigorously analyze the full tripartite entanglement of a mixed spin-(1/2,1) Heisenberg tetramer in the presence of an external magnetic field. The tetramer consists of two mixed spin-(1/2,1) dimers arranged to form a square plaquette, with distinct exchange couplings J and J1 between spins from the same and different dimers, respectively. The full tripartite entanglement is classified according to the tripartite negativity NABC defined as a geometric mean of all possible bipartite negativities NA|BC, NB|AC, and NC|AB, when the degrees of freedom of the fourth spin D of the tetrapartite system ABCD are traced out. We demonstrate that for J1/J>1, both trimeric subsystems 1/2-1-1 and 1/2-1/2-1 exhibit tripartite entanglement mediated through the central spin. This entanglement becomes more robust upon the application of an external magnetic field. The opposite limit J1/J<1 leads to the spontaneous emergence of full tripartite entanglement solely within the subsystem 1/2-1-1. However, the increasing magnetic field swiftly disrupts the three-particle correlation across the relevant ground-state phase boundary. Additionally, we identify that in the isotropic limit J1/J=1, the ground-state degeneracy significantly reduces the strength of full tripartite entanglement and favors nontrivial subtypes of full nonseparable states with none or just one spin-dimer entanglement. Finally, we discuss in detail the thermal stability of all fully tripartite nonseparable states.
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 J1. The degree of a genuine tripartite entanglement is evaluated according to the genuine tripartite negativity NABC defined as a geometric mean of all possible bipartite negativities corresponding to a decomposition into a single spin and the remaining spin dimer NA∣BC, NB∣AC and NC∣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 J1/J<1 becomes nonzero solely in the tripartite system 1/2−1−1 at low-enough magnetic fields. The opposite interaction limit J1/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.
We investigate key quantum characteristics of the mixed spin-(1/2,1,1/2) Heisenberg trimer under the influence of an external magnetic field. Specifically, we analyze the distributions of bipartite and tripartite entanglement quantified through the respective negativities and the l1-norm of coherence with the help of rigorous analytical and numerical methods. Our findings suggest that the heterotrinuclear molecular nanomagnet $[{{{\text{Cu}}^{2+}\text{L}}}_{2}{\text{Ni}}^{2+}{({\text{H}}_{2}\text{O})}_{2}]{({\text{ClO}}_{4})}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}3{\text{H}}_{2}\text{O}$, which represents an experimental realization of the mixed spin-(1/2,1,1/2) Heisenberg trimer, exhibits a significant bipartite entanglement between Cu2+ and Ni2+ magnetic ions along with robust tripartite entanglement among all three constituent Cu2+Ni2+Cu2+ magnetic ions. The significant bipartite and tripartite entanglement persists even at relatively high temperatures up to 37K and magnetic fields up to 46T, whereby the coherence is maintained even at elevated temperatures. It is evidenced that the aforementioned molecular complex with the magnetic core Cu2+Ni2+Cu2+ provides an intriguing quantum resource, which exhibits a star-shaped state within the singlet eigenstate at low magnetic fields and W-like state within the triplet eigenstate at moderate magnetic fields.
Mixed spin-(1/2,1/2,1) trimer with two different Landé g factors and two different exchange couplings is considered. The main feature of the model is nonconserving magnetization. The Hamiltonian of the system is diagonalized analytically. We presented a detailed analysis of the ground-state properties, revealing several possible ground-state phase diagrams and magnetization profiles. The main focus is on how nonconserving magnetization affects quantum entanglement. We have found that nonconserving magnetization can bring a continuous dependence of the entanglement quantifying parameter (negativity) on the magnetic field within the same eigenstate, while for the case of uniform g factors it is a constant. The main result is an essential enhancement of the entanglement in the case of uniform couplings for one pair of spins caused by an arbitrary small difference in the values of g factors. This enhancement is robust and brings almost a sevenfold increase of the negativity. We have also found a weakening of entanglement for other cases. Thus, nonconserving magnetization offers a broad opportunity to manipulate the entanglement by means of a magnetic field.
Heisenberg model spin systems offer favourable and manageable physical settings for generating and manipulating entangled quantum states. In this work mixed spin-(1/2,1/2,1) Heisenberg spin trimmer with two different but isotropic Landé g-factors and two different exchange constants is considered. The study undertakes the task of finding the optimal parameters to create entangled states and control them by external magnetic field. The primary objective of this work is to examine the tripartite entanglement of a system and the dependence of the tripartite entanglement on various system parameters. Particularly, the effects of non-conserving magnetization are in the focus of our research. The source of non-commutativity between the magnetic moment operator and the Hamiltonian is non-uniformity of g-factors. To quantify the tripartite entanglement, an entanglement measure refered to as “tripartite negativity” has been used in this work.