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Kramers theorem rely on odd total number of electrons. In reality, total number of electrons is about 10^23. Can those electrons be so smart to count the total number precisely and decide to form Kramers doublets or not?

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I think Kramers theorem is not really useful for a bulk material, as the concept of degeneracy fails to have a meaning. In a bulk material you will have energy bands, not discrete levels as in individual atoms. The doublets can still exist, but only if the electrons do not interact. –  Alexander Oct 16 '12 at 8:18

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Remember that for crystalline materials, we usually assume an infinite number of particles, and that electrons do not interact. This allows us to Fourier transform and see that each pseudo-momentum $k$ is independent --- essentially to consider a single unit cell. In this context, Kramer's theorem states that if there is an odd number of electrons per unit cell (we ignore proton and neutrons if we don't care about hyperfine structure; otherwise we would), and assuming time reversal invariance, there is (at least) a two-fold degeneracy of all energy levels. Indeed, this may be seen as the basis of topological insulators.

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I think Kramer's theorem is really only useful when you can write down a wavefunction for your system. It then tells you the degeneracy of the ground state of your wavefunction. If you have 1 mole of your material you couldn't write down a wavefunction for it so the degeneracy of the ground state would be meaningless.

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