It should be noted that in solids, not just spatial antisymmetry, but also symmetry can give rise to spin alignment. In particular, in many ferrites, there is an opposite spin coupling between the iron atoms and the oxygen ones. If two iron atoms are opposite in spin to the same oxygen atom, it implies that they must have aligned spins even if their electrons do not interact directly.
It comes as somewhat a surprise to discover that in this time of high-temperature superconductors, the mechanism of plain old ferromagnetism is still not understood that well if the magnetic material is a conductor, such as a piece of iron.
For a conductor, the description of the exclusion effect should really be at least partly in terms of band theory, rather than electrons localized at atoms. More specifically, Aharoni [2, p. 48] notes “There is thus no doubt in anybody’s mind that neither the itinerant electron theory nor the localized electron one can be considered to be a complete picture of the physical reality, and that they both should be combined into one theory.”
Sproull notes that in solid iron, most of the 4s electrons move to the 4d bands. That reduces the magnetization by reducing the number of unpaired electrons.
While Sproull [42, p. 282] in 1956 describes ferromagnetism as an interaction between electrons localized at neighboring atoms, Feynman [22, p. 37-2] in 1965 notes that calculations using such a model produce the wrong sign for the interaction. According to Feynman, the interaction is thought to occur with [4s] conduction band electrons acting as intermediaries. More recently, Aharoni [2, p. 44] notes: “It used to be stated [...] that nobody has been able to compute a positive exchange integral for Fe, and a negative one for Cu [...]. More modern computations [...] already have the right sign, but the magnitude of the computed exchange still differs considerably from the experimental value. Improving the techniques [...] keeps improving the results, but not sufficiently yet.”
Batista, Bonča, and Gubernatis note that “After seven decades of intense effort we still do not know what is the minimal model of itinerant ferromagnetism and, more importantly, the basic mechanism of ordering.” (Phys Rev Let 88, 2002, 187203-1) and “Even though the transition metals are the most well studied itinerant ferromagnets, the ultimate reason for the stabilization of the FM phase is still unknown.” (Phys Rev B 68, 2003, 214430-11)