The dispersion of ferromagnetic spin wave at low energy is $E\propto k^2$, while $E\propto k$ for antiferromagnetic case. Is there a simple/physical argument (such as symmetry) for these results?
Moreover, in a generic system with spontaneous continuous symmetry breaking, what determines the value of $n$ in the low energy dispersion $E\propto k^n$ of the Goldstone mode?
In the Wikipedia, the last paragraph of theory of spin wave states that the difference between the dispersion relation of phonon ($E\propto k$) and ferromagnetic magnon ($E\propto k^2$) results from the time reversal symmetry breaking of the ferromagnetic state. I'm confused about it since antiferromagnetic state also breaks time reversal symmetry but possesses dispersion $E\propto k$ as phonon. Can someone explain this statement further please?