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Molecules

Magnetic-Field-Orientation Dependent Thermal Entanglement of a Spin-1 Heisenberg Dimer: The Case Study of Dinuclear Nickel Complex with an Uniaxial Single-Ion Anisotropy

Azadeh Ghannadan, Jozef Strečka

Published Jun 5, 2021• 20 citations
Publisher

Abstract

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.

DOI
10.3390/molecules26113420

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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)22·2H2O 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.
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.
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.
The quantum and thermal entanglement in the mixed spin-(1/2,SSS) Heisenberg dimer with an uniaxial single-ion anisotropy
The concept of negativity is used to study the quantum and thermal entanglement in a generalized spin-(1/2,SSS) Heisenberg dimer under presence of an external magnetic field. The competition effect of interplay between the spin diversity, XXZ exchange as well as uniaxial single-ion anisotropy is additionally analysed with a goal to tune the degree and thermal stability of the pairwise entanglement. Obtained analytical results favour the antiferromagnetic spin-(1/2,SSS) Heisenberg dimer instead of the ferromagnetic one, where the higher degree of entanglement and higher threshold temperature are achieved at the same set of model parameters. It is demonstrated and analytically proven that the increasing magnitude of spin SSS in dimer with an easy-axis uniaxial single-ion anisotropy can enhance not only the thermal stability but simultaneously the degree of entanglement. It is shown additionally, that the further enhancement of entanglement can be improved in dimer with a half-odd-integer spin SSS as a consequence of different magnetic ground-states. In such system the low-enough thermal negativity saturates in its maximal value regardless the magnitude of half-odd-integer spin SSS. Because of the influence of the magnetic-field induced consecutive second-order phase transitions in dimer with S ⁣> ⁣1S\!>\!1S>1, the surprising oscillating behaviour of negativity upon the magnetic field is observed at low-enough temperature.
We investigate various types of quantum entanglement in the octanuclear heterometallic 3d/4f3d/4f3d/4f complexes denoted as Ni42+Gd43+{\mathrm{Ni}}_{4}^{2+}{\mathrm{Gd}}_{4}^{3+}Ni42+​Gd43+​ under an external magnetic field, using the exact diagonalization approach. These molecular magnets, which can be effectively described by Heisenberg spin models, consist of two identical {Ni22+Gd23+{\mathrm{Ni}}_{2}^{2+}{\mathrm{Gd}}_{2}^{3+}Ni22+​Gd23+​} cubane subunits bridged by acetate and hydroxide ligands. Our analysis reveals that their magnetization exhibits intermediate plateaus at low temperatures, indicating distinct ground states characteristic of Ni-containing compounds. Using negativity as a measure of quantum entanglement, we examine the influence of single-ion anisotropy and magnetic field on tetrapartite, bipartite, 1--3 tangle, and 2--2 tangle entanglements in two families of Ni42+Gd43+{\mathrm{Ni}}_{4}^{2+}{\mathrm{Gd}}_{4}^{3+}Ni42+​Gd43+​ complexes: (1) without anisotropy and (2) with anisotropy. Complex (1) exhibits strong bipartite entanglement between Ni ions, which persists up to $T\ensuremath{\approx}3.0\phantom{\rule{0.28em}{0ex}}\text{K}$ and $B\ensuremath{\approx}4.0\phantom{\rule{0.28em}{0ex}}\text{T}$, but shows significantly weaker tetrapartite entanglement and vanishing bipartite entanglement between $\mathrm{Gd}\ensuremath{\cdots}\mathrm{Gd}$ and $\mathrm{Ni}\ensuremath{\cdots}\mathrm{Gd}$ pairs. In contrast, complex (2) displays nonzero and sizable values for all types of entanglement considered. These findings emphasis the crucial role of single-ion anisotropy in generating and shaping the entanglement landscape of heterometallic Ni42+Gd43+{\mathrm{Ni}}_{4}^{2+}{\mathrm{Gd}}_{4}^{3+}Ni42+​Gd43+​ complexes. Notably, we find that the 1--3 tangle entanglement between a Ni ion and the remaining sites in a cubane unit serves as a reliable indicator of ground-state phase transitions, exhibiting distinct changes across phase boundaries irrespective of the presence of single-ion anisotropy.