Which mechanism causes ferromagnetism in iron? There are at least three different mechanisms which can give rise to ferromagnetic order in iron. 


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*First is due to the band electrons called band magnetism or itinerant magnetism which is an exchange interaction between conduction electrons. 


The page 91 of Fundamentals of Many-body physics by W. Nolting, page 251 (sidenote 1) of Oxford Solid State Basics by Steven H. Simons tells that Fe is itinerant.


*Second is indirect exchange i.e. exchange between unpaired d electrons and conduction electrons.

*The third is the direct exchange between localized magnetic moments of two neighbouring Fe ions as described by the Heisenberg model. 
Which one of them is responsible for ferromagnetism in iron (and also cobalt and nickel) and why? I expect that the third effect would be least because d orbitals are inner orbitals and do not have much overlap.
I read this, this, this and the question titled "What is the difference between a localized and itinerant magnetism?". None seem to address my concern. 
 A: A lot of work has been done since the 1984 review by Moriya and Takahashi, both theory and experiments.
On the theoretical side, calculations have been refined a lot. The effect of the on-site $dd$ correlation energy $U$ was taken into account first by LSDA+U, then by LSD+DMFT (a dynamic mean field theory), for example this paper from 2001 about high temperature: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.87.067205
On the experimental side, instead of studying the spectra of the paramagnetic phase by heating up the sample, techniques were developed to just heat the electrons with ultra-fast laser pulses. For example this paper from 2017 by Eich:
https://advances.sciencemag.org/content/advances/3/3/e1602094.full.pdf
Figure 1B shows what would be expected in the photoemission spectrum contrasting the case of an itinerant Stoner picture and the case of a Heisenberg-like picture with local moments. The data support the latter.
A: For ferromagnetic materials like iron the cause of alignment of the individual atomic magnetic moments is the direct exchange interaction, that is number 3. This is the mainstream explanation as given in the Feynman lectures ( https://en.m.wikipedia.org/wiki/Ferromagnetism). 
Fe is definitely not an itinerant magnet. The electrons responsible for the magnetism are localised. The magnetic transition is an order-disorder transition. In itinerant magnets delivalised electrons are responsible and the transition is a phase transition. No magnetism persists above the transition temperature, apart of course from the lo al moments. Note that the d electrons form filled bands of one spin direction. A full band Slater determinant can be written equivalently in local and in crystal orbitals. This fact may be relevant to the debate.
A: Normally, it is 1. This is when the unpaired outer valence electrons of neighboring atoms overlap and the distribution of their electric charge in space are farther apart when they have parallel spins. This reduces the electrostatic energy so when they have parallel spin, it is more stable.
Normally this is very short ranged, called intra-atomic (electrons in the same atom) exchange or direct exchange between neighboring atoms.
But iron has a characteristic, it is able to create something called magnetic domains. Though the spins are aligned inside the domains, the domains themselves cancel out each other and the whole peace of iron will not be magnetic.
Now this is when longer ranged interactions can occur via intermediary atoms, called superexchange, or indirect exchange, that is 2 and 3.
Please see here:
https://arxiv.org/ftp/cond-mat/papers/0701/0701423.pdf
