We study the dynamics of nearest-neighbor entanglement for a one-dimensional spin chain with a nearest-neighbor time-dependent Heisenberg coupling $J\left(t\right)$ between the spins in the presence of a time-dependent external magnetic field $h\left(t\right)$ 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 $J\left(t\right)$ 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 $J\left(t\right)=\lambda h\left(t\right)$, for constant $\lambda$, exactly, we showed that the time evolution and asymptotic value of entanglement are dictated solely by the parameter $\lambda =J/h$ 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.

• Received 18 April 2011

DOI:https://doi.org/10.1103/PhysRevA.84.022314