  Ferromagnetism is typically discussed in terms of the exchange interaction
and magnetic anisotropies. Yet real samples are inevitably affected by the
magnetostatic dipole-dipole interaction. Because of this interaction, a theorem
(R.B. Griffiths, Free Energy of interacting magnetic dipoles, Phys. Rev. 176,
655 (1968)) forbids a spontaneous magnetization in, nota bene,
three-dimensional bodies. Here we discuss perpendicularly and in-plane
magnetized ferromagnetic bodies in the shape of a slab of finite thickness. In
perpendicularly magnetized slabs, magnetic domains are energetically favored
when the lateral size is sufficiently large, i.e. there is no spontaneous
magnetization. For in-plane magnetization, instead, spontaneous magnetization
is possible below a critical thickness which, in real thin films, could be as
small as few monolayers. At this critical thickness, we predict a genuine phase
transition to a multi-domain state. These results have implications for
two-dimensional ferromagnetism.
