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