  The possibility to induce long range ferromagnetic order by doping oxides
with transition metal ions has become a very exciting challenge in the last
decade. Theoretically, it has been claimed that Mn doped ZrO$_2$ could be a
very promising spintronic candidate and that high critical temperatures could
be already achieved even for a low Mn concentration. Some experiments have
reported room temperature ferromagnetism (RT-FM) whilst some others only
paramagnetism. When observed, the nature of RT-FM appears to be controversial
and not clearly understood. In this study, we propose to clarify and shed light
on some of theses existing issues. A detailed study of the critical
temperatures and low energy magnetic excitations in Mn doped ZrO$_2$ is
performed. We show that the Curie temperatures were largely overestimated
previously, due to the inadequate treatment of both thermal and transverse
fluctuations, and disorder. It appears that the Mn-Mn couplings can not explain
the observed RT-FM. We argue, that this can be attributed to the interaction
between large moments induced in the vicinity of the manganese. This is similar
to the non-magnetic defect induced ferromagnetism reported in oxides,
semiconductors and graphene/graphite.
