  We present a study of magnetic field induced quantum phase transitions in
insulating systems. A generalized scaling theory is used to obtain the
temperature dependence of several physical quantities along the quantum
critical trajectory ($H=H_{C}$, $T\to0$) where $H$ is a longitudinal external
magnetic field and $H_{C}$ the critical value at which the transition occurs.
We consider transitions from a spin liquid at a critical field $H_{C1}$ and
from a fully polarized paramagnet, at $H_{C2}$, into phases with long range
order in the transverse components. The transitions at $H_{C1}$ and $H_{C2}$
can be viewed as Bose-Einstein condensations of magnons which however belong to
different universality classes since they have different values of the dynamic
critical exponent $z$. Finally, we use that the magnetic susceptibility is an
entanglement witness to discuss how this type of correlation sets in as the
system approaches the quantum critical point along the critical trajectory,
$H=H_{C2}$, $T\to0$.
