  When the density of a nuclear system is decreased, homogeneous states undergo
the so-called Mott transition towards clusterised states, e.g. alpha
clustering, both in nuclei and in nuclear matter. Here we investigate such a
quantum phase transition (QPT) by using microscopic energy density functional
(EDF) calculations both with the relativistic and the Gogny approaches on the
diluted $^{16}$O nucleus. The evolution of the corresponding single-particle
spectrum under dilution is studied, and a Mott-like transition is predicted at
about 1/3 of the saturation density. Complementary approaches are used in order
to understand this QPT. A study of spatial localisation properties as a
function of the density allows to derive a value of the Mott density in
agreement with the one obtained by fully microscopic calculations in $^{16}$O
and in nuclear matter. Moreover a study of the spontaneous symmetry breaking of
the rotational group in $^{16}$O, down to the discrete tetrahedral one,
provides further insight on the features displayed by the single-particle
spectrum obtained within the EDF approach.The content of the tetrahedrally
deformed A-nucleon product state in terms of spherical particle-hole
configurations is investigated. Finally a study of quartet condensation and the
corresponding macroscopic QPT is undertaken in infinite matter.
