  Below the critical temperature $T_c\simeq 600 K$, an iso-structural
transition, named $\gamma$-$\alpha$ transition,can be induced in Cerium by
applying pressure. This transition is first-order, and is accompanied by a
sizable volume collapse. A conclusive theoretical explanation of this
intriguing phenomenon has still not been achieved, and the physical pictures
proposed so far are still under debate. In this work, we illustrate
zero-temperature first-principle calculations which clearly demonstrate that
the $\gamma$-$\alpha$ transition is induced by the interplay between the
electron-electron Coulomb interaction and the spin-orbit coupling. We address
the still unresolved problem on the existence of a second low-$T$ critical
point, i.e., whether the energetic effects alone are sufficient or not to
induce the $\gamma$-$\alpha$ transition at zero temperature.
