# How to understand the difference of spin wave excitation for ferromagnetism (quadratic dispersion) and anti-ferromagnetism (linear dispersion)?

As we know, the dispersion of spin excitation (magnon/spin wave) for ferromagnetic(FM) system is quadratic as $$k\rightarrow 0$$, but is linear for anti-ferromagnetic(AFM) system as $$k\rightarrow 0$$. I am confused about the reason for this difference. In other words, what is the physical meaning of "linear" dispersion and "quadratic" dispersion?

From my point of view, it seems that quadratic dispersion comes from "quantum fluctuation", i.e. FM has no zero point fluctuation, but if we add some other terms which introduce zero point fluctuation, the excitation dispersion will change to linear. Is my argument reasonable? And I still cannot even understand the relation between quantum fluctuation and linear dispersion.

Also, as we know, the effective description for AFM fluctuation without long-range order is (non-linear sigma model): $$E=|k| \longrightarrow E=\sqrt{k^2+\Delta}$$ where $$\Delta$$ is related with inverse of correlation length. Thus, I am confused about effective description for FM fluctuation without long-range order? What's the form of dispersion for it?

There is a classical picture of spin waves in ferromagnets that answers your first question. Think first of a single spin in an external magnetic field, say along the $$z$$-direction. We know what happens: the spin will undergo Larmor precession, in which the $$S_{x,y}$$ components of the spin oscillate with a $$\pi/2$$ phase shift. We cannot choose initial conditions to make $$S_{x,y}$$ oscillate independently of each other because they are canonically conjugate. Accordingly, Larmor precession is described by first-order dynamics.
• @MerlinZhang I had implicitly in mind Lagrangian field theory for the Goldstone bosons of the broken symmetry. A term of the type $\phi\partial_t\chi$ in the Lagrangian makes $\phi$ and $\chi$ canonically conjugate, and at the same time makes the equation of motion of first order in time derivatives. If you are familiar with the effective Lagrangian technique, you will find inspirehep.net/literature/1092745 very readable. The issue of canonical conjugation is discussed around Eq. (12). The idea originally goes back to Nambu. Commented May 13, 2020 at 7:28