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International Journal of Quantum Information

Entanglement in a spin ring with anisotropic interactions

Marcos E. Gaudiano, Omar Osenda

Published Oct 1, 2015• 3 citations
Publisher

Abstract

The relationship between entanglement and anisotropy is studied in small spin chains with periodic boundary conditions. The Hamiltonian of the spin chains is given by a slight modification of the dipolar Hamiltonian. The effect of the anisotropy is analyzed using the concurrence shared by spin pairs, but the study is not restricted to nearest-neighbor (NN) entanglement. It is shown that, under rather general conditions, the inclusion of anisotropic terms diminishes the entanglement shared between the spins of the chain irrespective of its range or its magnetic character.

DOI
10.1142/s0219749915500574

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The relationship between entanglement and anisotropy is studied in small spin chains with periodic boundary conditions. The Hamiltonian of the spin chains is given by a slight modification of the dipolar Hamiltonian. The effect of the anisotropy is analyzed using the concurrence shared by spin pairs, but the study is not restricted to nearest-neighbor (NN) entanglement. It is shown that, under rather general conditions, the inclusion of anisotropic terms diminishes the entanglement shared between the spins of the chain irrespective of its range or its magnetic character.
The entanglement in an anisotropic spin-1 Heisenberg chain with a uniform magnetic field is investigated. The ground-state entanglement will undergo two different kinds of transitions when the anisotropy Δ and the amplitude of the magnetic field B are varied. The thermal entanglement of the nearest neighbour always declines when B increases no matter what the value of the anisotropy is. It is very interesting to note that the entanglement of the next-nearest neighbour can increase to a maximum at a certain magnetic field. Regardless of the boundary condition, the nearest-neighbour entanglement always decreases and approaches to a constant value when the size of the system is very large. The constant value of open boundary condition is much larger than that of periodic boundary condition.
We use a disordered antiferromagnetic spin-1∕21∕21∕2 chain with anisotropic exchange coupling to model an array of interacting qubits. All qubits have the same level spacing, except two, which are called the defects of the chain. The level spacings of the defects are equal and much larger than all the others. We investigate how the entanglement between the two defects depends on the anisotropy of the system. When the anisotropy coupling is much larger than the energy difference between a defect and an ordinary qubit, the two defects become strongly entangled. Small anisotropies, on the contrary, may decrease the entanglement, which is, in this case, also much affected by the number of excitations. The analysis is made for nearest-neighbor and next-nearest-neighbor defects. The decrease in the entanglement for nearest neighbor defects is not very significant, especially in large chains.
The pair-wise thermal entanglement in a four-qubit Heisenberg XXZ chain is investigated to study the role of anisotropy when an external magnetic field is included. It is found that pair-wise entanglement is absent between nearest- and next-nearest neighbouring qubits with anisotropic parameter Δ⩽−1. For two nearest-neighbouring qubits, increasing the parameter can not only induce the entanglement, but also extend the entanglement region in terms of magnetic field B and temperature T . For two next-nearest-neighbouring qubits, increasing anisotropic parameter can shift the location of the entanglement and control the extent of the entanglement in terms of magnetic field at a finite temperature.
We study the entanglement in the one-dimensional Kondo necklace model with exact diagonalization, calculating the concurrence as a function of the Kondo coupling J and an anisotropy η in the interaction between conduction spins, and we review some results previously obtained in the limiting cases η = 0 and 1. We observe that as J increases, localized and conduction spins get more entangled, while neighboring conduction spins diminish their concurrence; localized spins require a minimum concurrence between conduction spins to be entangled. The anisotropy η diminishes the entanglement for neighboring spins when it increases, driving the system to the Ising limit η = 1 where conduction spins are not entangled. We observe that the concurrence does not give information about the quantum phase transition in the anisotropic Kondo necklace model (between a Kondo singlet and an antiferromagnetic state), but calculating the von Neumann block entropy with the density matrix renormalization group in a chain of 100 sites for the Ising limit indicates that this quantity is useful for locating the quantum critical point.
The entanglement resonance in anisotropic spin-1/2 Heisenberg chains of different couplings is investigated when the nearest neighbor coupling is periodically modulated with external magnetic field. When the modulation frequency equals twice of the magnetic field, the entanglement resonance is larger than that at other modulation frequencies and decreases as the number of spins in the chain increases. When the modulation frequency equals the magnetic field, the entanglement resonance can be reduced to a quite low value by varying the coupling along z axis.