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Journal of Experimental and Theoretical Physics Letters

Effect of anisotropy on the entanglement of quantum states in a spin chain

Petr F. Kartsev, V. А. Kashurnikov

Published Sep 1, 2004• 2 citations
Publisher

Abstract

The effect of the anisotropy of the interaction of a spin chain in the XXZ Heisenberg model on the concurrence of the states of neighboring sites is studied. When anisotropy increases, the maximum concurrence in a magnetic field increases above the value reached in the absence of the field. The maximum magnetic field allowing entanglement is linearly related to the anisotropy parameter.

DOI
10.1134/1.1830665

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The effect of the anisotropy of the interaction of a spin chain in the XXZ Heisenberg model on the concurrence of the states of neighboring sites is studied. When anisotropy increases, the maximum concurrence in a magnetic field increases above the value reached in the absence of the field. The maximum magnetic field allowing entanglement is linearly related to the anisotropy parameter.
The effects of spin-spin interaction on thermal entanglement of a two-qubit Heisenberg XYZ model with different inhomogeneous magnetic fields are investigated. It is shown that the entanglement is dependent on the spin-spin interaction and the inhomogeneous magnetic fields. The larger the J i (i-axis spin-spin interaction), the higher critical value the B i (i-axis uniform magnetic field) has. Moreover, in the weak-field regime, the larger J i corresponds to more entanglement, while in the strong-field regime, different J i correspond to the same entanglement. In addition, it is found that with the increase of J i , the concurrence can approach the maximum value more rapidly for the smaller B i , and can reach a larger value for the smaller b i (i-axis nonuniform magnetic field). So we can get more entanglement by increasing the spin-spin interaction J i , or by decreasing the uniform magnetic field B i and the nonuniform magnetic field b i .
We derive the analytic expression of the concurrence in the quantum Heisenberg XYZ model and discuss the influence of parameters J , Δ and Γ on the concurrence. By choosing different values of Γ and Δ, we obtain the XX , XY , XXX and XXZ chains. The concurrence decreases with increasing temperature. When T →0, the concurrence reaches its maximum value 1, i.e. the entangled state, |Ψ⟩ = (((2) 1/2 )/2)(|01⟩-|10⟩), is maximum entanglement. For the XXZ chain, when Γ→∞, the concurrence will meet its maximum value 𝒞 max = (sinh (1/ T ))/(cosh (1/ T )).
Antiferromagnetic spin chains play an important role in condensed matter physics and statistical mechanics. Recently XXX spin chain was discussed in relation to the information theory. We consider here localizable entanglement, introduced recently by F.Verstraete, M.Popp and J.I.Cirac. That is how much entanglement can be localized on two spins on average by performing local measurements on the other individual spins in a system of many interacting spins. We consider the ground state in antiferromagnetic spin chains and study localizable entanglement between two spins as a function of the distance. We start with isotropic spin chain. Then we study effect of anisotropy and magnetic field. We conclude that anisotropy increases localizable entanglement. We found an explicit dependence of critical exponents in XXZ spin chain on magnetic field. We discovered that the cases of high symmetry corresponds to high sensitivity of magnetic field. We also calculated the concurrence before the measurement to illustrate that the measurment raises the concurrence.
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.
Antiferromagnetic spin chains play an important role in condensed matter and statistical mechanics. Recently XXX\mathit{XXX}XXX spin chain was discussed in relation to information theory. Here we consider localizable entanglement. It is how much entanglement can be localized on two spins by performing local measurements on other individual spins (in a system of many interacting spins). We consider the ground state of antiferromagnetic spin chain. We study localizable entanglement [represented by concurrence] between two spins. It is a function of the distance. We start with isotropic spin chain. Then we study effects of anisotropy and magnetic field. We conclude that anisotropy increases the localizable entanglement. We discovered high sensitivity to a magnetic field in cases of high symmetry. We also evaluated concurrence of these two spins before the measurement to illustrate that the measurement raises the concurrence.