The entanglement dynamics between two end spins in an anisotropic spin-1/2 Heisenberg chain is investigated when the external magnetic field and the nearest-neighbor coupling are periodically modulated.The entanglement resonance occurs at certain modulation frequency and magnitude of the external magnetic filed.The maximum entanglement appears only at some specific frequencies while it can keep the maximum for quite broad range of the magnitude.Thus,the entanglement can be controlled by the external magnetic field.
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 study the dynamics of entanglement for a one-dimensional spin system, where spins are coupled through nearest-neighbor exchange interaction and subject to different external magnetic fields. First we examine the system size effect on the entanglement with three different external magnetic fields changing with time
t: an exponential function ${e}^{[\ensuremath{-}Kt]}$ and two periodic
sin[Kt] and
cos[Kt] functions, where
K is a control parameter. We have found that the entanglement fluctuates shortly after a disturbance by an external magnetic field when the system size is small. For larger system size, the entanglement reaches a stable state for a long time before it fluctuates. However, this fluctuation of entanglement disappears when the system size goes to infinity. We also show that in a periodic external magnetic field, the nearest-neighbor entanglement displays a periodic structure with a period related to that of the magnetic field. Moreover, changing the direction of the magnetic field will destroy the concurrence in the system.
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 .
The thermal entanglement in the Heisenberg
XYZ chain is investigated in the presence of an external magnetic field B. In the two-qubit system, the critical magnetic field
Bc is increased by introducing the interaction of the z component of two neighboring spins
Jz. This interaction not only improves the critical temperature
Tc, but also enhances the entanglement for a particular fixed B. We also analyze the pairwise entanglement between nearest neighbors in three qubits. The pairwise entanglement, for a fixed T, can become strong by controlling B and
Jz.We study the dynamics of nearest-neighbor entanglement for a one-dimensional spin chain with a nearest-neighbor time-dependent Heisenberg coupling
J(t) between the spins in the presence of a time-dependent external magnetic field
h(t) at zero and finite temperatures. We consider different forms of time dependence for the coupling and magnetic field: exponential, hyperbolic, and periodic. Solving the system numerically, we examined the system-size effect on the entanglement asymptotic value. It was found that, for a small system size, the entanglement starts to fluctuate within a short period of time after applying the time-dependent coupling. The period of time increases as the system size increases and disappears completely as the size goes to infinity. Testing the effect of the transition constant for an exponential or hyperbolic coupling showed a direct impact on the asymptotic value of the entanglement; the larger the constant is, the lower the asymptotic value and the more rapid decay of entanglement are, which confirms the nonergodic character of the system. We also found that, when
J(t) is periodic, the entanglement shows a periodic behavior with the same period, which disappears upon applying periodic magnetic field with the same frequency. Solving the case $J(t)=\ensuremath{\lambda}h(t)$, for constant $\ensuremath{\lambda}$, exactly, we showed that the time evolution and asymptotic value of entanglement are dictated solely by the parameter $\ensuremath{\lambda}=J/h$ rather than the individual values of
J and
h, not only when they are time independent and at zero temperature, but also when they are time dependent but proportional at zero and finite temperatures for all degrees of anisotropy.