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arXiv:cond-mat/0503170 · Physical Review A

Thermal entanglement properties of small spin clusters

Indrani Bose, Amit Tribedi

Published Aug 15, 2005• 57 citations
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Abstract

Exchange interactions in spin systems can give rise to quantum entanglement in the ground and thermal states of the systems. We consider a spin tetramer, with spins of magnitude $\frac{1}{2}$, in which the spins interact via nearest-neighbor, diagonal, and four-spin interactions of strength ${J}_{1}$, ${J}_{2}$, and $K$, respectively. The ground- and thermal-state entanglement properties of the tetramer are calculated analytically in the various limiting cases. Both bipartite and multipartite entanglements are considered and a signature of the quantum phase transition (QPT), in terms of the entanglement ratio, is identified. The first-order QPT is accompanied by discontinuities in the nearest-neighbor and diagonal concurrences. The magnetic properties of a $S=\frac{1}{2}$ antiferromagnetic polyoxovanadate compound V12 are well explained by tetramers, with ${J}_{2}=0$, $K=0$, in which the spins interact via the isotropic Heisenberg exchange interaction Hamiltonian. Treating the magnetic susceptibility $\ensuremath{\chi}$ as an entanglement witness (EW), an estimate of the lower bound of the critical entanglement temperature ${T}_{c}$ below which entanglement is present in the experimental compound, is determined. Two other cases considered include the symmetric tetramer---i.e., tetrahedron $({J}_{1}={J}_{2},K=0)$---and the symmetric trimer. In both the cases, there is no entanglement between a pair of spins in the thermal state but multipartite entanglement is present. A second EW based on energy provides an estimate of the entanglement temperature ${T}_{E}$ below which the thermal state is definitely entangled. This EW detects bipartite entanglement in the case of the tetramer describing a square of spins (the case of V12 ) and multipartite entanglement in the cases of the tetrahedron and symmetric trimer.

DOI
10.1103/physreva.72.022314

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Exchange interactions in spin systems can give rise to quantum entanglement in the ground and thermal states of the systems. We consider a spin tetramer, with spins of magnitude 12\frac{1}{2}21​, in which the spins interact via nearest-neighbor, diagonal, and four-spin interactions of strength J1{J}_{1}J1​, J2{J}_{2}J2​, and KKK, respectively. The ground- and thermal-state entanglement properties of the tetramer are calculated analytically in the various limiting cases. Both bipartite and multipartite entanglements are considered and a signature of the quantum phase transition (QPT), in terms of the entanglement ratio, is identified. The first-order QPT is accompanied by discontinuities in the nearest-neighbor and diagonal concurrences. The magnetic properties of a S=12S=\frac{1}{2}S=21​ antiferromagnetic polyoxovanadate compound V12 are well explained by tetramers, with J2=0{J}_{2}=0J2​=0, K=0K=0K=0, in which the spins interact via the isotropic Heisenberg exchange interaction Hamiltonian. Treating the magnetic susceptibility \ensuremathχ\ensuremath{\chi}\ensuremathχ as an entanglement witness (EW), an estimate of the lower bound of the critical entanglement temperature Tc{T}_{c}Tc​ below which entanglement is present in the experimental compound, is determined. Two other cases considered include the symmetric tetramer---i.e., tetrahedron (J1=J2,K=0)({J}_{1}={J}_{2},K=0)(J1​=J2​,K=0)---and the symmetric trimer. In both the cases, there is no entanglement between a pair of spins in the thermal state but multipartite entanglement is present. A second EW based on energy provides an estimate of the entanglement temperature TE{T}_{E}TE​ below which the thermal state is definitely entangled. This EW detects bipartite entanglement in the case of the tetramer describing a square of spins (the case of V12 ) and multipartite entanglement in the cases of the tetrahedron and symmetric trimer.
In the context of spin systems, exchange interactions play a crucial role in generating quantum entanglement within the ground and thermal states. This entanglement has been identified as a valuable reserve for Quantum Information and Computation protocols. In our study, we focus on a spin tetramer composed of spins with a magnitude of 1/2. These spins interact with each other through pairwise nearest-neighbour and diagonal interactions denoted as J1 and J2, respectively, as well as arm-wise and diagonal-wise 4-spin interactions of strengths K1 and K2 respectively . The geometric arrangement of the tetramer leads to frustration – a phenomenon observed in physical systems. By performing analytical calculations, we explore the quantum correlation properties of the tetramer in both its ground and thermal states. Our investigation includes the study of bipartite and multipartite entanglements, Quantum Discord and Fidelity. Notably, we identify signatures of quantum critical transition lines, which can be referred as the ‘entanglement transition lines’ signalled differently by different measures.
The entanglement quantum properties of a spin-1/2 Ising-Heisenberg model on a symmetrical diamond chain were analyzed. Due to the separable nature of the Ising-type exchange interactions between neighboring Heisenberg dimers, calculation of the entanglement can be performed exactly for each individual dimer. Pairwise thermal entanglement was studied in terms of the isotropic Ising-Heisenberg model and analytical expressions for the concurrence (as a measure of bipartite entanglement) were obtained. The effects of external magnetic field H and next-nearest neighbor interaction J(m) between nodal Ising sites were considered. The ground state structure and entanglement properties of the system were studied in a wide range of coupling constant values. Various regimes with different values of ground state entanglement were revealed, depending on the relation between competing interaction strengths. Finally, some novel effects, such as the two-peak behavior of concurrence versus temperature and coexistence of phases with different values of magnetic entanglement, were observed.
Quantum discord (QD) is a general measure of bipartite quantum correlations with a potential role in quantum information processing tasks. Spin clusters serve as ideal candidates for the implementation of some of the associated protocols. In this paper, we consider a symmetric spin trimer and a tetramer which describe a number of known molecular magnets and compute the QD in the ground and thermal states of the clusters. The variations of the QD as a function of an anisotropy parameter, magnetic field and temperature are investigated. We obtain a number of interesting results such as a finite value of the QD in the trimer ground state for which the pairwise entanglement is known to be zero, differences in the nature of some of the variations in the ferromagnetic and antiferromagnetic cases, and discontinuous jumps in the magnitude of the QD at first-order quantum phase transition points. A remarkable feature that is observed is that the QD completely vanishes only in the asymptotic limit of temperature T → ∞. We further study the dynamics of the QD and the pairwise entanglement at T = 0 under the effect of a dephasing channel describing the interaction of the reduced spin cluster state with independent local environments. The QD is found to vanish asymptotically as t → ∞. In the case of the spin trimer, the pairwise entanglement has a zero value at all times and reaches a zero value in a finite time in the case of the tetramer.
In this paper, we study three specific aspects of entanglement in small spin clusters. We first study the effect of inhomogeneous exchange coupling strengths on the entanglement properties of the S=12S=\frac{1}{2}S=21​ antiferromagnetic linear chain tetramer compound NaCuAsO4\mathrm{Na}\mathrm{Cu}\mathrm{As}{\mathrm{O}}_{4}NaCuAsO4​. The entanglement gap temperature TE{T}_{E}TE​ is found to have a nonmonotonic dependence on the value of $\ensuremath{\alpha}$, the exchange coupling inhomogeneity parameter. We next determine the variation of TE{T}_{E}TE​ as a function of SSS for a spin dimer, a trimer, and a tetrahedron. The temperature TE{T}_{E}TE​ is found to increase as a function of SSS but the scaled entanglement gap temperature tE{t}_{E}tE​ goes to zero as SSS becomes large. Last, we study a spin-1 dimer compound to illustrate the quantum complementarity relation. We show that in the experimentally realizable parameter region, magnetization and entanglement plateaus appear simultaneously at low temperatures as a function of the magnetic field. Also, the sharp increase in one quantity as a function of the magnetic field is accompanied by a sharp decrease in the other so that the quantum complementarity relation is not violated.
Quantum discord (QD) is a general measure of bipartite quantum correlations with a potential role in quantum information processing tasks. Spin clusters serve as ideal candidates for the implementation of some of the associated protocols. In this paper, we consider a symmetric spin trimer and a tetramer which describe a number of known molecular magnets and compute the QD in the ground and thermal states of the clusters. The variations of the QD as a function of an anisotropy parameter, magnetic field and temperature are investigated. We obtain a number of interesting results such as a finite value of the QD in the trimer ground state for which the pairwise entanglement is known to be zero, differences in the nature of some of the variations in the ferromagnetic and antiferromagnetic cases, and discontinuous jumps in the magnitude of the QD at first-order quantum phase transition points. A remarkable feature that is observed is that the QD completely vanishes only in the asymptotic limit of temperature T → ∞. We further study the dynamics of the QD and the pairwise entanglement at T = 0 under the effect of a dephasing channel describing the interaction of the reduced spin cluster state with independent local environments. The QD is found to vanish asymptotically as t → ∞. In the case of the spin trimer, the pairwise entanglement has a zero value at all times and reaches a zero value in a finite time in the case of the tetramer.
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.