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Magnetochemistry

On the Concurrent Bipartite Entanglement of a Spin-1 Heisenberg Diamond Cluster Developed for Tetranuclear Nickel Complexes

Azadeh Ghannadan, Katarína Karľová, Jozef Strečka

Published Nov 12, 2022• 9 citations
Publisher

Abstract

The bipartite entanglement of a quantum spin-1 Heisenberg diamond cluster in the presence of the external magnetic field is quantified through the negativity, which is calculated for spin pairs from a diagonal and a side of the diamond spin cluster taking into consideration two different coupling constants. The magnetic field may cause a few crossings of energy levels of the spin-1 Heisenberg diamond cluster, which is responsible at low enough temperatures for a stepwise dependence of the negativity on the magnetic field accompanied with a drop of the negativity at respective magnetic-field-driven transitions due to emergence of mixed states. It is shown that the bipartite entanglement between spin pairs on a diagonal and a side of the diamond spin cluster is concurrent although they may eventually become both nonzero albeit not fully saturated. It is predicted that the tetranuclear nickel complex [Ni4(μ-CO3)2(aetpy)8](ClO4)4 (aetpy = 2-aminoethyl-pyridine), which represents an experimental realization of the spin-1 Heisenberg diamond cluster with two different antiferromagnetic coupling constants, exhibits a substantial bipartite entanglement between two spin-1 Ni2+ magnetic ions from a shorter diagonal of the diamond spin cluster up to temperatures approximately about 50 K and up to magnetic fields about 70 T.

DOI
10.3390/magnetochemistry8110156

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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.

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The bipartite entanglement of a quantum spin-1 Heisenberg diamond cluster in the presence of the external magnetic field is quantified through the negativity, which is calculated for spin pairs from a diagonal and a side of the diamond spin cluster taking into consideration two different coupling constants. The magnetic field may cause a few crossings of energy levels of the spin-1 Heisenberg diamond cluster, which is responsible at low enough temperatures for a stepwise dependence of the negativity on the magnetic field accompanied with a drop of the negativity at respective magnetic-field-driven transitions due to emergence of mixed states. It is shown that the bipartite entanglement between spin pairs on a diagonal and a side of the diamond spin cluster is concurrent although they may eventually become both nonzero albeit not fully saturated. It is predicted that the tetranuclear nickel complex Ni4(μ-CO3)2(aetpy)84 (aetpy = 2-aminoethyl-pyridine), which represents an experimental realization of the spin-1 Heisenberg diamond cluster with two different antiferromagnetic coupling constants, exhibits a substantial bipartite entanglement between two spin-1 Ni2+ magnetic ions from a shorter diagonal of the diamond spin cluster up to temperatures approximately about 50 K and up to magnetic fields about 70 T.
The bipartite entanglement in pure and mixed states of a quantum spin-1 Heisenberg dimer with exchange and uniaxial single-ion anisotropies is quantified through the negativity in a presence of the external magnetic field. At zero temperature the negativity shows a marked stepwise dependence on a magnetic field with two abrupt jumps and plateaus, which can be attributed to the quantum antiferromagnetic and quantum ferrimagnetic ground states. The magnetic-field-driven phase transition between the quantum antiferromagnetic and quantum ferrimagnetic ground states manifests itself at nonzero temperatures by a local minimum of the negativity, which is followed by a peculiar field-induced rise of the negativity observable in a range of moderately strong magnetic fields. The rising temperature generally smears out abrupt jumps and plateaus of the negativity, which cannot be distinguished in the relevant dependencies above a certain temperature. It is shown that the thermal entanglement is most persistent against rising temperature at the magnetic field, for which an energy gap between a ground state and a first excited state is highest. Besides, temperature variations of the negativity of the spin-1 Heisenberg dimer with an easy-axis single-ion anisotropy may exhibit a singular point-kink, at which the negativity has discontinuity in its first derivative. The homodinuclear nickel complex [Ni2(Medpt)2(μ-ox)(H2O)2](ClO4)2·2H2O provides a suitable experimental platform of the antiferromagnetic spin-1 Heisenberg dimer, which allowed us to estimate a strength of the bipartite entanglement between two exchange-coupled Ni2+ magnetic ions on the grounds of the interaction constants reported previously from the fitting procedure of the magnetization data. It is verified that the negativity of this dinuclear compound is highly magnetic-field-orientation dependent due to presence of a relatively strong uniaxial single-ion anisotropy.
Bipartite quantum and thermal entanglement is quantified within pure and mixed states of a mixed spin-(1/2,11/2,11/2,1) Heisenberg dimer with the help of negativity. It is shown that the negativity, which may serve as a measure of the bipartite entanglement at zero as well as nonzero temperatures, strongly depends on intrinsic parameters---for instance, exchange and uniaxial single-ion anisotropy---in addition to extrinsic parameters such as temperature and magnetic field. It turns out that a rising magnetic field unexpectedly reinforces the bipartite entanglement due to the Zeeman splitting of energy levels, which lifts the twofold degeneracy of the quantum ferrimagnetic ground state. The maximal bipartite entanglement is thus reached within a quantum ferrimagnetic phase at sufficiently low but nonzero magnetic fields under the assumption that the gyromagnetic ggg factors of the spin-1/2 and spin-1 magnetic ions are equal and the uniaxial single-ion anisotropy is half of the exchange constant. It is suggested that the heterodinuclear complex $[\mathrm{Ni}(\mathrm{dpt})({\mathrm{H}}_{2}\mathrm{O})\mathrm{Cu}(\mathrm{pba})]\ifmmode\cdot\else\textperiodcentered\fi{}2{\mathrm{H}}_{2}\mathrm{O}$ [pba = 1,3-propylenebis(oxamato) and dpt = bis-(3-aminopropyl)amine], which affords an experimental realization of the mixed spin-(1/2,11/2,11/2,1) Heisenberg dimer, remains strongly entangled up to relatively high temperatures (about 115 K) and magnetic fields (about 140 T) that are comparable with the relevant exchange constant.
The concept of negativity is adapted in order to explore the quantum and thermal entanglement of the mixed spin-(1/2,S) Heisenberg dimers in presence of an external magnetic field. The mutual interplay between the spin size S, XXZ exchange and uniaxial single-ion anisotropy is thoroughly examined with a goal to tune the degree and thermal stability of the pairwise entanglement. It turns out that the antiferromagnetic spin-(1/2,S) Heisenberg dimers exhibit higher degree of entanglement and higher threshold temperature in comparison with their ferromagnetic counterparts when assuming the same set of model parameters. The increasing spin magnitude S accompanied with an easy-plane uniaxial single-ion anisotropy can enhance not only the thermal stability but simultaneously the degree of entanglement. It is additionally shown that the further enhancement of a bipartite entanglement can be achieved in the mixed spin-(1/2,S) Heisenberg dimers, involving half-odd-integer spins S. Under this condition the thermal negativity saturates at low-enough temperatures in its maximal value regardless of the magnitude of half-odd-integer spin S. The magnetic field induces consecutive discontinuous phase transitions in the mixed spin-(1/2,S) Heisenberg dimers with S>1, which are manifested in a surprising oscillating magnetic-field dependence of the negativity observed at low enough temperature.
Bipartite entanglement is comprehensively investigated in the mononuclear molecular complex (Et3NH)[Ni(hfac)2L], where HL denotes 2-(2-hydroxy-3-methoxy-5-nitrophenyl)-4,4,5,5-tetramethyl-4,5-dihydro-1H-imidazol-3-oxide-1-oxyl and hfacH stands for hexafluoroacetylacetone. From the magnetic point of view, the molecular compound (Et3NH)[Ni(hfac)2L] consists of an exchange-coupled spin-1 Ni2+ magnetic ion and a spin-12 nitronyl-nitroxide radical substituted nitrophenol. The nickel-radical molecular complex affords an experimental realization of a mixed spin-(12, 1) Heisenberg dimer with a strong antiferromagnetic exchange coupling, J/kB = 505 K, and two distinct g-factors, gRad = 2.005 and gNi = 2.275. By adopting this set of magnetic parameters, we demonstrate that the Zeeman splitting of a quantum ferrimagnetic ground-state doublet due to a weak magnetic field may substantially reinforce the strength of bipartite entanglement at low temperatures. The molecular compound (Et3NH)[Ni(hfac)2L] maintains sufficiently strong thermal entanglement, even at room temperature, vanishing only above 546 K. Specifically, the thermal entanglement in the nickel-radical molecular complex retains approximately 40% of the maximum value, corresponding to perfectly entangled Bell states at room temperature, which implies that this magnetic compound provides a suitable platform of a molecular qubit with potential implications for room-temperature quantum computation and quantum information processing.
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 JJJ and J1J_1J1​. The degree of a genuine tripartite entanglement is evaluated according to the genuine tripartite negativity NABC{\cal N}_{ABC}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{\cal N}_{A|BC}NA∣BC​, NB∣AC{\cal N}_{B|AC}NB∣AC​ and NC∣AB{\cal N}_{C|AB}NC∣AB​ after degrees of freedom of the last fourth spin DDD 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 ⁣− ⁣11/2\!-\!1\!-\!11/2−1−1 and 1/2 ⁣− ⁣1/2 ⁣− ⁣11/2\!-\!1/2\!-\!11/2−1/2−1 were identified. It was found that the genuine tripartite negativity for the interaction ratio J1/J ⁣< ⁣1J_1/J\!<\!1J1​/J<1 becomes nonzero solely in the tripartite system 1/2 ⁣− ⁣1 ⁣− ⁣11/2\!-\!1\!-\!11/2−1−1 at low-enough magnetic fields. The opposite interaction limit J1/J ⁣> ⁣1J_1/J\!>\!1J1​/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.