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In ferromagnetism the spins align in the same direction, but in antiferromagnetism the spins align in the opposite direction. Both are due to the so-called 'exchange interaction', with a negative exchange constant ($J$) corresponding to ferromagnetism and a positive exchange constant corresponding to antiferromagnetism.

However, I don't understand how the exchange interaction can lead to antiferromagnetism. What I learnt from Griffiths's intro to quantum, is that due to the exchange interaction i.e. exchange force, the expectation value between two identical fermions is larger than the expectation value between two distinguishable particles, which lowers the Coulomb interaction energy (assuming we are talking about 2 electrons here). Therefore, the spins want to align since this is more energetically favorable. What is the conceptual mechanism behind antiferromagnetism then?

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  • $\begingroup$ Just continue onto a solid state physics textbook, and you will learn to do it. It is just a matter of considering more cases. The case covered in Griffiths is just one of many possibilities. $\endgroup$
    – naturallyInconsistent
    Commented Apr 19 at 11:53
  • $\begingroup$ I don't understand why it is a matter of 'cases'. What Griffiths describes seems to imply something very general: electrons want to align their spin in order to minimize the Coulomb interaction energy. I am asking for general and conceptual explanations. Perhaps you can refer me to a specific section where they explain this specifically? Because I wasn't able to find it. $\endgroup$
    – Stallmp
    Commented Apr 19 at 12:03

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The exchange interaction gives a beautiful picture of the Heisenberg Hamiltonian where the exchange is only the coupling in the ground state. If you want to understand the origin of this coupling you should rely on the coulomb interaction and the Hubbard picture. Normally the translational symmetry and the coulomb interaction are in competition. When the translational operator is enhanced and the solid is a Bloch-like, the coulomb interaction defines the ferromagnetic coupling when in this case the coulomb interaction are strong enough, the triplet states are dominant for reducing the translation or the hopping. On the other hand, when the translational operator is diminished and the coulomb interaction are important, the antiferromagnetic coupling for enhancing this translation and the hopping in this Wannier-like solid.

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