  We discuss the quantum dynamics of an isolated composite system consisting of
weakly interacting many-body subsystems. We focus on one of the subsystems, S,
and study the dependence of its quantum correlations and decoherence rate on
the state of the weakly-coupled complementary part E, which represents the
environment. As a theoretical laboratory, we consider a composite system made
of two stacked quantum Ising chains, locally and homogeneously weakly coupled.
One of the chains is identified with the subsystem S under scrutiny, and the
other one with the environment E. We investigate the behavior of S at
equilibrium, when the global system is in its ground state, and under
out-of-equilibrium conditions, when the global system evolves unitarily after a
soft quench of the coupling between S and E. When S develops quantum critical
correlations in the weak-coupling regime, the associated scaling behavior
crucially depends on the quantum state of E whether it is characterized by
short-range correlations (analogous to those characterizing disordered phases
in closed systems), algebraically decaying correlations (typical of critical
systems), or long-range correlations (typical of magnetized ordered phases). In
particular, different scaling behaviors, depending on the state of E, are
observed for the decoherence of the subsystem S, as demonstrated by the
different power-law divergences of the decoherence susceptibility that
quantifies the sensitivity of the coherence to the interaction with E.
