  A thermodynamic limit chain of spin-1/2 particles with XX and three-spin
interactions (TSI) is considered. Using the fermionization technique, the
Hamiltonian of the chain with periodic boundary conditions is exactly
diagonalized. In the ground-state phase diagram of the chain system, a quantum
critical point separates the Luttinger liquid and chiral phases. Selecting
one-spin as an open quantum system, the rest of the spins play the role of its
environment. By choosing different initial states, we studied the dynamics of
the entanglement between the open quantum system and its environment. In the
initial state, the state of the open quantum system is a superposition of up
and down states and the environment is polarized. The revival of the
entanglement is observed in the whole range of interactions as an indicator of
non-Markovian dynamics. Our exact results revealed that non-Markovian behavior
is independent of the initial state. By tuning the initial state, the open
quantum system will be completely entangled with the environment at a special
time called entanglement-time, tE. The value of the mentioned entanglement-time
decreased by increasing TSI. The signature of a quantum critical point is
confirmed by investigating the long-time average of the entanglement and using
the trace-distance as a witness of non-Markovianity.
