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.
Abstract This work considers the spin-1/2 Heisenberg dimer, the mixed spin-(1/2,1) Heisenberg dimer and their quantum entanglement, particularly, the dependence of the entanglement on the system characteristics and the external magnetic field. The quantum entanglement in this work is determined using two different methods, namely ‘Negativity’ and ‘Concurrence’. The dependence of the corresponding quantities on spins Landé g-factors, system anisotropy constants and exchange constant is discussed.
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.
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.
The spatial distribution of entanglement within a spin-1/2 Heisenberg star composed from a single central spin and three peripheral spins is examined in the presence of an external magnetic field using the Kambe projection method, which allows an exact calculation of the bipartite and tripartite negativity serving as a measure of the bipartite and tripartite entanglement. Apart from a fully separable polarized ground state emergent at high-enough magnetic fields, the spin-1/2 Heisenberg star exhibits at lower magnetic fields three outstanding nonseparable ground states. The first quantum ground state exhibits the bipartite and tripartite entanglement over all possible decompositions of the spin star into any pair or triad of spins, whereby the bipartite and tripartite entanglement between the central and peripheral spins dominates over that between the peripheral spins. The second quantum ground state has a remarkably strong tripartite entanglement between any triad of spins in spite of the lack of bipartite entanglement. The central spin of the spin star is separable from the remaining three peripheral spins within the third quantum ground state, where the peripheral spins are subject to the strongest tripartite entanglement arising from a two-fold degenerate W-state.