  We show how the phenomenon of factorization in a quantum many body system is
of collective nature. To this aim we study the quantum discord $Q$ in the one
dimensional XY model in a transverse field. We analyze the behavior of $Q$ at
both the critical point and at the non critical factorizing field. The
factorization is found to be governed by an exponential scaling law for $Q$. We
also address the thermal effects fanning out from the anomalies occurring at
zero temperature. Close to the quantum phase transition, $Q$ exhibits a
finite-temperature crossover with universal scaling behavior, while the
factorization phenomenon results in a non trivial pattern of correlations
present at low temperature.
