  The mechanism of how critical end points of the first-order valence
transitions (FOVT) are controlled by a magnetic field is discussed. We
demonstrate that the critical temperature is suppressed to be a quantum
critical point (QCP) by a magnetic field. This results explain the field
dependence of the isostructural FOVT observed in Ce metal and YbInCu_4.
Magnetic field scan can lead to reenter in a critical valence fluctuation
region. Even in the intermediate-valence materials, the QCP is induced by
applying a magnetic field, at which the magnetic susceptibility also diverges.
The driving force of the field-induced QCP is shown to be a cooperative
phenomenon of the Zeeman effect and the Kondo effect, which creates a distinct
energy scale from the Kondo temperature. The key concept is that the closeness
to the QCP of the FOVT is capital in understanding Ce- and Yb-based heavy
fermions. It explains the peculiar magnetic and transport responses in CeYIn_5
(Y=Ir, Rh) and metamagnetic transition in YbXCu_4 for X=In as well as the sharp
contrast between X=Ag and Cd.
