This problem is difficult to discuss briefly and in its generality. For this reason, for a comprehensive overview of the subject matter I refer you to the excellent book Lecture Notes on Electron Correlation and Magnetism, by Patrik Fazekas (World Scientific, Singapore, 2003). Check in particular 'antifrromagnetism' in the index of this book.

One case that is relatively easily discussed and immediately appreciated, concerns that of a half-filled band of conduction electrons on a bipartite lattice. In this case, because of perfect nesting (where a subset of non-zero measure of the points on the non-interacting Fermi surface are connected to each other by some of  the reciprocal-lattice vectors of the underlying lattice), the static susceptibility of the non-interacting (metallic) state becomes logarithmically divergent at wave-vectors coinciding with the underlying reciprocal lattice vectors, implying that even a weak interaction is capable of turning the ground state of the system into an anti-ferromagnetic insulating state. There is an element of self-consistency involved here, since for an insulating state, where the Fermi energy is located inside the lowest-lying excitation gap, the density of electronic states at zero temperature is identically vanishing at the Fermi energy, resulting in a bounded static susceptibility at the above-mentioned reciprocal lattice vectors.

Away from half-filling of the above-mentioned conduction band (on a bipartite lattice), where the condition of perfect nesting does not apply, the situation is very complex and the physical understanding should relay on non-trivial theoretical and computational calculations. Often variational calculations (here commonly using the Gutzwiller variational wave function) can greatly help the understanding of the coincidence of antiferromagnetism and the insulating state of the condensed-matter systems.

Ps. See also:

[1] P.W. Anderson, Antiferromagnetism. Theory of Superexchange Interaction, Phys. Rev. 79, 350 (1950).

[2] P.W. Anderson, New Approach to the Theory of Superexchange, Phys. Rev. 115,  2 (1959).

   Dear Lei,

   Although the relation antiferromagnetism-insulator has a difficult explanation if we want to give a general answer and the reference of the Facekas is a very good one, there is a simple explanation for special cases, which show clearly the Physics involved in your question.

   Taking a Hubbard model at half-filling lattice, it is simple to obtain that it is equivalent to a Heisenberg model where the coupling constant J is negative and with value

                     J=-4t2/U

being t the hopping of the electrons and U the Coulomb repulsion in the singlet. While the ferromagnetic triplet state has one positive coupling and energy.

   Thus this is equivalent to say that the electrons (or holes) close to the Fermi level have to have much less kinetic energy for the antiferromagnetic (degenerate singlets) than to the ferromagnetic order (triplet states). In any case the antiferromagnetism helps to decrease the electronic conductivity and the Neel temperature is propotional to

             TN ~ t exp[-(t/U)1/2]

 for the increasing part of the antiferromagnetic phase and

             TN ~ t2/U

 for the decreasing part of the this magnetic phase. This doesn't proves that a gap (of the real bands) is created in the case of antiferromagnetism, which is not so simple to show it, but the ingredients of the transport associated are clearly seen.

    I hope that this can helps.

     Dear Lei,

     If you want more details you can go to my contributions where you have papers on this issue and one of them is :

     https://www.researchgate.net/publication/1836932_On_the_absence_of_conduction_electrons_in_the_antiferromag

Article On the absence of conduction electrons in the antiferromagne...