Ground-state properties of antiferromagnetic Heisenberg spin rings

Bärwinkel, Klaus; Schmidt, Heinz‐Jürgen; Schnack, Jürgen

doi:10.1016/s0304-8853(00)00481-9

Cited by 44 publications

(49 citation statements)

References 29 publications

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“…However, in contrast to the triangles and the related lattices, the frustration in odd AFM wheels is not due to a direct competition of two exchange bonds but the incompatibility of the closed boundary conditions with a bipartite sublattice structure. It also cannot be captured by a tripartite sublattice structure, which allowed to successfully describe trianglebased frustrated magnetic molecules such as the Keplerate Fe 30 . 29 The lack of partitedness at any spatial range puts the odd AFM wheels into an interesting class of materials by itself.…”

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confidence: 99%

Inelastic neutron scattering studies on the odd-membered antiferromagnetic wheel Cr8Ni

et al. 2012

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18 ], or Cr 8 Ni, is a prominent example of an odd-membered antiferromagnetic "wheel." A detailed characterization of the magnetic properties of Cr 8 Ni has been conducted. Inelastic neutron scattering (INS) is used to investigate the energy and momentum transfer dependence of the low-lying spin excitations, including excited states inaccessible by other experimental techniques. The richness of the INS data, in conjunction with microscopic spin Hamiltonian simulations, enables an accurate characterization of the magnetic properties of Cr 8 Ni. Nearest-neighbor exchange constants of J CrCr = 1.31 meV and J CrNi = 3.22 meV are determined, and clear evidence of axial single-ion anisotropy is found. The parameters determined by INS are shown to fit magnetic susceptibility. The spectroscopic identification of several successive S = 1 excited total spin states and lowest spin band excitations show that the rotational band picture, valid for bipartite AFM wheels, breaks down for this odd-numbered wheel. The exchange constants determined here differ from previous efforts based on bulk measurements, and possible reasons are discussed. The large J CrNi /J CrCr ratio in Cr 8 Ni puts this wheel into a regime with strong quantum fluctuations in which the ground state can be described with a valence bond solid state picture.

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Section: Introductionmentioning

confidence: 99%

Inelastic neutron scattering studies on the odd-membered antiferromagnetic wheel Cr8Ni

et al. 2012

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“…For N ≤ 7, the isotropic Heisenberg Hamiltonian can be analytically solved [25,26]. Here, we give the analytical results of the eigenvalues of the anisotropic Heisenberg model with N = 5, without use of Bethe's Ansatz, from which the analytical expressions for entanglement and mixedness of two nearest-neighbor qubits are readily obtained.…”

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Analytical results for entanglement in the five-qubit anisotropic Heisenberg model

Wang

2004

Physics Letters A

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We solve the eigenvalue problem of the five-qubit anisotropic Heisenberg model, without use of Bethe's Ansatz, and give analytical results for entanglement and mixedness of two nearest-neighbor qubits. The entanglement takes its maximum at ∆ = 1 (∆ > 1) for the case of zero (finite) temperature with ∆ being the anisotropic parameter. In contrast, the mixedness takes its minimum at ∆ = 1 (∆ > 1) for the case of zero (finite) temperature.PACS numbers: 03.65. Ud, Recently, the study of entanglement properties of many-body systems has received much attention [1]- [22]. To obtain analytical results for entanglement, one may consider the case of infinite lattice or a small lattice with a few qubits. It is hard to get some analytical results between these two extreme cases.It was well-known that the anisotropic Heisenberg model can be solved formally by Bethe's Ansatz method [23,24] for arbitrary number of qubits N , however, we have to solve a set of transcendental equations. For N ≤ 7, the isotropic Heisenberg Hamiltonian can be analytically solved [25,26]. Here, we give the analytical results of the eigenvalues of the anisotropic Heisenberg model with N = 5, without use of Bethe's Ansatz, from which the analytical expressions for entanglement and mixedness of two nearest-neighbor qubits are readily obtained.It is interesting to see that the entanglement properties of a pair of nearest-neighbor qubits at a finite temperature is completely determined by the partition function. The entanglement, quantified by the concurrence [27], relates to the partition function Z via [28,29,30] being the internal energy, andbeing the correlation function. Here, β = 1/T and the Boltzmann's constant k = 1. Thus, once we know the eigenenergies versus the temperature and the anisotropic parameter, we can completely determine the entanglement.There exists another concept, the mixedness of a state, is central in quantum information theory [31]. For instance, Bose and Vedral have shown that entangled states become useless for quantum teleportation on exceeding a certain degree of mixedness [32]. Mixedness is also related to quantum entanglement. We will study both the entanglement and mixedness properties. Eigenvalue problem.The anisotropic Heisenberg Hamiltonian is given bywhereis the swap operator between qubit i and j, σ i = (σ ix , σ iy , σ iz ) is the vector of Pauli matrices, and J is the exchange constant. We have assumed the periodic boundary condition, i.e., N +1 ≡ 1. In the following discussions, we also assume J = 1 (antiferromagnetic case) and ∆ ≥ 0.Since we impose the periodic boundary condition, the Hamiltonian is translational invariant, i.e., [H, T ] = 0, where T is the cyclic right shift operator defined asThe translational invariant symmetry can be used to reduce the Hamiltonian matrix to smaller submatrices by a factor of N [33]. Now we focus our attention to five-qubit settings, and solve the eigenvalue problem of the anisotropic Heisenberg model. Since [H, J z ] = 0, the whole 32-dimentional Hilbert space can be divide...

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“…But now we know that it is valid for any number of qubits. From the available results [17][18][19][20] of the ground-state energy we can determine the concurrence directly. For the case of even-number qubits the concurrence is given in Ref.…”

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“…It is known that [19] the ground state which has S z = 0, is non-degenerate if N is even, and the ground state which has S z = ±1/2, is four-fold degenerate if N is odd. Then the ground state is a pure (mixed) state for even (odd) N .…”

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Threshold temperature for pairwise and many-particle thermal entanglement in the isotropic Heisenberg model

Wang

2002

Phys. Rev. A

145

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We study the threshold temperature for pairwise thermal entanglement in the spin-1/2 isotropic Heisenberg model up to 11 spins and find that the threshold temperature for odd and even number of qubits approaches the thermal dynamical limit from below and above, respectively. The threshold temperature in the thermodynamical limit is estimated. We investigate the many-particle entanglement in both ground states and thermal states of the system, and find that the thermal state in the four-qubit model is four-particle entangled before a threshold temperature.PACS numbers: 03.65. Ud, 75.10.Jm Entanglement is a nonlocal quantum correlation and entangled states constitute indeed a valuable resource in quantum information processing [1]. Entanglement in systems of interacting spins [2-6] as well as in systems of indistinguishable particles [7][8][9][10][11] has been investigated. In particular entanglement in both the ground state [2,3] and thermal state [4-6] of a spin-1/2 Heisenberg spin chain have been analyzed in the literature. The intriguing issue of the relation between entanglement and quantum phase transition [12] have been addressed in a few quite recent papers [13,14].The state we will consider is the thermal equilibrium state in a canonical ensemble. The system state at finite temperature is given by the Gibb's density operator ρ T = exp (−H/kT ) /Z, where Z =tr[exp (−H/kT )] is the partition function, H the system Hamiltonian, k is Boltzmann's constant which we henceforth will take equal to 1, and T the temperature. As ρ T represents a thermal state, the entanglement in the state is called thermal entanglement [4]. It is important to stress that, although the central object of statistical physics, the partition function, is determined by the eigenvalues of H only, thermal entanglement properties generally require in addition the knowledge of the energy eigenstates. In some models such as the isotropic Heisenberg qubit rings the pairwise thermal entanglement can be completely determined by the partition function [15].Let us briefly review the main results of Ref.[15]. We consider a physical model of a ring of N qubits interacting via the isotropic Heisenberg Hamiltonianwhere S j,j+1 = 1 2 (1 + σ i · σ i+1 ) is the swap operator between qubit i and j, σ i = (σ ix , σ iy , σ iz ) is the vector of Pauli matrices, and J is the exchange constant. In Ref.[15] we proved that there is no thermal entanglement between two nearest-neighbour qubits in the ferromagnetic (J < 0) isotropic Heisenberg rings at any temperature. So we only consider the antiferromagnetic case and the exchange constant J is then assumed to be 1. A direct relation is established between the concurrence [16] quantifying two-qubit entanglement and a macroscopic thermodynamical function, the internal energy U [15]:where β = 1/T and C(N ) refers to the concurrence for two nearest-neighbor qubits of the N -qubit ring. The entanglement is uniquely determined by the partition function of the system. Note that there is a slight difference between the...

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Ground-state properties of antiferromagnetic Heisenberg spin rings

Cited by 44 publications

References 29 publications

Inelastic neutron scattering studies on the odd-membered antiferromagnetic wheel Cr8Ni

Inelastic neutron scattering studies on the odd-membered antiferromagnetic wheel Cr8Ni

Analytical results for entanglement in the five-qubit anisotropic Heisenberg model

Threshold temperature for pairwise and many-particle thermal entanglement in the isotropic Heisenberg model

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