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From what I can tell only certain materials appear as ferromagnets or antiferromagnets (which one depends on their exchange interactions). For these materials if risen above a certain temperature, the Currie temperature in the case of ferro and Neel in the case of antiferro, they lose their ordered magnetic properties and become paramagnets.

This is to do with the strength of the thermal fluctuations. If we take an arbitrary material (e.g. Cu or Si) I would except what when dropped below a low enough temperature they would start the see these ordering effects due to interactions. Is this the case? i.e can any material be made into a (anti)ferromagnet at low enough temperatures? Is so why and if not why not?

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First of all you should consider what drives the (anti)ferromagnetism – unpaired dipoles (e.g. electrons) which align spontaneously. These are called local moments, and there is a bunch of literature regarding how local moments are formed.

If you have a material with no free electrons (i.e. all atomic orbitals are filled), you would not be able to obtain anti/ferromagnetic ordering.

There is a second mechanism which prohibits anti/ferromagnetic ordering which is called frustration. Consider for instance ising spins (i.e. up or down) on a triangular lattice – it will be impossible for you to align the spins in such a way that all of them are pointing in the opposite direction on each bond. There are other mechanisms that induce frustration (for instance the Kitaev Honeycomb Model). This phenomenon of frustration leads to the fact that even down to 0 K, you don't get magnetic ordering. Frustrated magnets harbor so called spin liquids with only short range interactions and all kinds of exotic excitations.

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