  We study the time evolution of entanglement of two spins in anisotropically
coupled quantum dot interacting with the unpolarized nuclear spins environment.
We assume that the exchange coupling strength in the z-direction $J_z$ is
different from the lateral one $J_l$. We observe that the entanglement decays
as a result of the coupling to the nuclear environment and reaches a saturation
value, which depends on the value of the exchange interaction difference $J=\|
J_l-J_z\|$ between the two spins and the strength of the applied external
magnetic field. We find that the entanglement exhibits a critical behavior
controlled by the competition between the exchange interaction $J$ and the
external magnetic field. The entanglement shows a quasi-symmetric behavior
above and below a critical value of the exchange interaction. It becomes more
symmetric as the external magnetic field increases. The entanglement reaches a
large saturation value, close to unity, when the exchange interaction is far
above or below its critical value and a small one as it closely approaches the
critical value. Furthermore, we find that the decay rate profile of
entanglement is linear when the exchange interaction is much higher or lower
than the critical value but converts to a power law and finally to a Gaussian
as the critical value is approached from both directions. The dynamics of
entanglement is found to be independent of the exchange interaction for
isotropically coupled quantum dot.
