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?

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.