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What is the explanation for total spin conservation forbidding the spin wave gap in Heisenberg magnets?

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"Total spin conservation" means global $SU(2)$ spin-rotation symmetry (a continuous symmetry) of the Heisenberg model, and "spin wave" indicates an ordered ground state that spontaneously breaks the spin-rotation symmetry.

Thus, according to Goldstone theorem, there must be a gapless mode for spin wave.

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    $\begingroup$ Thanks. Sorry for jabber. How about taking Dyzaloshinskii-Moriya interaction or any other anisotropicity into account? Invalidate Goldstone theorem? $\endgroup$ – xiaohuamao Dec 17 '13 at 13:30
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    $\begingroup$ @huotuichang Yes. "Goldstone theorem" is about the spontaneous symmetry breaking. If we add some spin-rotation symmetry broken terms(anisotropicity) to the Heisenberg model, then the symmetry is explicitly broken at the Hamiltonian level and there would be no spontaneous symmetry breaking, thus we can not use the "Goldstone theorem" in this case. $\endgroup$ – Kai Li Dec 17 '13 at 15:31

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