  We present a characterization of quantum phase transitions in terms of the
the overlap function between two ground states obtained for two different
values of external parameters. On the examples of the Dicke and XY models, we
show that the regions of criticality of a system are marked by the extremal
points of the overlap and functions closely related to it. Further, we discuss
the connections between this approach and the Anderson orthogonality
catastrophe as well as with the dynamical study of the Loschmidt echo for
critical systems.
