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The Physics of Metals and Metallography

Effect of anisotropy in XXZ Heisenberg chain on entanglement properties

Petr F. Kartsev

Published Nov 28, 2006• 0 citations
Publisher

Abstract

We study the entanglement of Heisenberg XXZ spin chain using the exact diagonalization method. Increasing parameter of anisotropy makes it possible to increase concurrence, although its maximum value is reached at a nonzero external field. A linear relation between the maximum value of the magnetic field allowing entanglement and the parameter of anisotropy has been established.

DOI
10.1134/s0031918x06140201

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We study the entanglement of Heisenberg XXZ spin chain using the exact diagonalization method. Increasing parameter of anisotropy makes it possible to increase concurrence, although its maximum value is reached at a nonzero external field. A linear relation between the maximum value of the magnetic field allowing entanglement and the parameter of anisotropy has been established.
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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 )).
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.
In this paper, the entanglement in a spin-1 Heisenberg XXZ chain with uniform and nonuniform external magnetic field is investigated in the terms of the measure of entanglement called 'negativity'. The effects of the external magnetic field, the anisotropy parameter and the temperature on the negativity are discussed respectively. It can be seen that the entanglement in this spin chain for any temperature is symmetric with respect to zero magnetic field. We calculate the partially transposed density matrix and give qualitatively results at different temperatures. In order to show the results clearly, the behavior of negativity is presented for five different cases. It can be found that the temperature and the external magnetic field can effect the feature of the thermal entanglement significantly. In addition, under certain condition, the negativity may be enhanced under a nonuniform magnetic field.
We study the entanglement of a two-qubit one-dimensional XYZ Heisenberg chain in thermal equilibrium at temperature T. We obtain an analytical expression for the concurrence of this system in terms of the parameters of the Hamiltonian and T. We show that depending on the relation among the coupling constants, it is possible to increase the amount of entanglement of the system by increasing its anisotropy. We also show numerically that for all sets of the coupling constants entanglement is a monotonically decreasing function of the temperature T, proving that we must have at least an external magnetic field in the z-direction to obtain a behavior where entanglement increases with T.