Posted by Steve Simon on April 15, 2014, 7:17 pm, in reply to "Antiferromagnetism and the mean field theory"
One points up one points down! If you didn't keep track of them separately they would have to point the same direction.
Admittedly by symmetry in zero external magnetic field, by symmetry, you will find that S_A = -S_B, so that you really don't have to think about them differently as long as you keep track of the minus sign.
However, if you add an external magnetic field as well, this then breaks the symmetry and S_A is not -S_B, and you really do have to keep track of two different quantities.
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