Why does total spin conservation law forbid the spin wave gap in Heisenberg magnets?

Question

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

This post imported from StackExchange Physics at 2014-08-22 05:06 (UCT), posted by SE-user huotuichang

Answer 1

"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.

This post imported from StackExchange Physics at 2014-08-22 05:06 (UCT), posted by SE-user K-boy

Comments

Thanks. Sorry for jabber. How about taking Dyzaloshinskii-Moriya interaction or any other anisotropicity into account? Invalidate Goldstone theorem?

This post imported from StackExchange Physics at 2014-08-22 05:06 (UCT), posted by SE-user huotuichang

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@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.

This post imported from StackExchange Physics at 2014-08-22 05:06 (UCT), posted by SE-user K-boy