  We show that bipartite entanglement in a one-dimensional quantum spin model
undergoing time-evolution under local Markovian environments can be frozen over
time. We demonstrate this by using a number of paradigmatic quantum spin models
in one dimension, including the anisotropic XY model in the presence of a
uniform and an alternating transverse magnetic field (ATXY), the XXZ model, the
XYZ model, and the $J_1-J_2$ model involving the next-nearest-neighbor
interactions. We show that the length of the freezing interval, for a chosen
pair of nearest-neighbor spins, may remain independent of the length of the
spin-chain, for example, in paramagnetic phases of the ATXY model, indicating a
scale-invariance. Such freezing of entanglement is found to be robust against a
change in the environment temperature, presence of disorder in the system, and
whether the noise is dissipative, or not dissipative. Moreover, we connect the
freezing of entanglement with the propagation of information through a quantum
many-body system, as considered in the Lieb-Robinson theorem. We demonstrate
that the variation of the freezing duration exhibits a quadratic behavior
against the distance of the nearest-neighbor spin-pair from the noise-source,
obtained from exact numerical simulations, in contrast to the linear one as
predicted by the Lieb-Robinson theorem.
