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We discuss that a low-energy effective Lagrangian relying on SO(3)$\rightarrow$ SO(2) is applicable for a ferrimagnet as well as a ferromagnetand an antiferromagnet. The analysis of the particle states shows that thereexist not only massless modes with the dispersion relations $\omega \propto|\bm{k}|,\, |\bm{k}|^2$, i.e., the so-called type-I and type-II Nambu-Goldstonemodes, respectively, but also gapped modes with $\omega \proptom^2+|\bm{k}|^2$. We clarify how the coefficients of the terms with one timederivative and those with two time derivatives in the effective Lagrangiandetermine the order parameters specifying whether the system is in aferromagnetic, antiferromagnetic or ferrimagnetic state; we stress that thegapped mode related to the spontaneous symmetry breaking appears only in theferrimagnetic system and not in the ferromagnetic and antiferromagneticsystems. We also establish the power counting scheme and calculate thescattering amplitudes and thereby the scattering lengths between the twoNambu-Goldstone bosons. We show that the scattering length of the gapped modeis finite and proportional to the gap. This characteristic property of thegapped NG mode can be used to discriminate it from gapped excitations whichoriginate in other mechanisms. Finally, we study the effects of the explicitsymmetry breaking that are given by an external magnetic field and a single-ionanisotropy, and show that the external magnetic fields do not have any effectson the scattering amplitudes in all the spin systems as was known for theferromagnet system. In contrast, the anisotropy does affect the scatteringamplitudes, the phase shift, and the scattering length except for spin 1/2systems. This result supports the possibility of the Efimov effect in spinsystems discussed in previous studies.

Effective field theory and the scattering process for magnons in the ferromagnet, antiferromagnet, and ferrimagnet