  At the heart of current information nanotechnology lies the search for ideal
platforms hosting the smallest possible magnets, i.e. single atoms with
magnetic moments pointing out-of-plane, as requested in a binary-type of
memory. For this purpose, a 2D material such as graphene would be an ideal
substrate thanks to its intrinsic low electron and phonon densities, as well as
its 6-fold symmetry. Here we investigate, from first-principles, a fundamental
mechanism detrimental for the magnetic stability: the zero-point
spin-fluctuations modifying the effective energy landscape perceived by the
local spin moments of 3$d$ and 4$d$ transition metal atoms deposited on a free
standing graphene. Utilizing time-dependent density functional theory and by
virtue of the fluctuation-dissipation theorem, these spontaneous quantum
fluctuations are found to be negligible for most of the 3$d$ elements, in
strong contrast to the 4$d$ atoms. Surprisingly, we find that such fluctuations
can promote the magnetic stability by switching the easy direction of the
magnetic moment of Tc from being initially in-plane to out-of-plane. The
adatom-graphene complex gives rise to impurity states settling in some cases
the magnetocrystalline anisotropy energy --- the quantity that defines the
energy barrier protecting the magnetic moments and, consequently, the
spin-excitation behavior detectable with inelastic scanning tunneling
spectroscopy. A detailed analysis is provided on the impact of electron-hole
excitations, damping and lifetime of the spin-excitations on the dynamical
behavior of the adsorbed magnetic moments on graphene.
