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How can one understand the concept of Kramer's Degeneracy for an antiferromagnetic system where spin-up and spin-down bands overlap due to net zero magnetic moment?

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You may want to have a look at my answer to a related question. Basically, in the usual single-particle-model picture of an antiferromagnetic system, the total wavefunction is like $\Psi=|\uparrow\downarrow\uparrow\downarrow\uparrow\downarrow\cdots\rangle$. Its time-reversed version, $\tilde{\Psi}=|\downarrow\uparrow\downarrow\uparrow\downarrow\uparrow\cdots\rangle$, amounts to the system where every spin of the antiferromagnetic system is flipped, which certainly has the same energy as $\Psi$. Therefore the Kramers degeneracy is present, but only on a global level.

Note that the energy band model is an approximation, and is basically a single particle model. So while the Kramers theorem always holds when you flip all the spins of the system at once, it may or may not hold when you flip only one of them.

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