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arXiv:2311.17444

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

Hana Vargová, Jozef Strečka

Published Nov 29, 2023• 0 citations
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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.

DOI
10.48550/arxiv.2311.17444

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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 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.
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 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.
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.
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.
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.
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.