Hamiltonian formalism for nonlinear spin wave dynamics under antisymmetric interactions: Application to Dzyaloshinskii-Moriya interaction

A Hamiltonian formalism is applied for the investigation of nonlinear spin wave dynamics under the influence of antisymmetric magnetic interactions. In the framework of this formalism we account not only for symmetric magnetic interactions (exchange, dipole-dipole, magnetocrystalline anisotropy), but also for antisymmetric interactions, such as Dzyaloshinskii-Moriya exchange interaction. The account of antisymmetric exchange, in general, could lead to the appearance of an additive nonreciprocal term in the spin wave dispersion law. We present the generalization of the linear transformation for the diagonalization of quadratic part of the Hamiltonian (so-called ``third Holstein-Primakoff transformation'') for the antisymmetric case, which allowed us to obtain generalized expressions for the coefficients of the nonlinear three- and four-magnon interactions. As an example, nonlinear spin wave interactions in ultrathin ferromagnetic nanowires and films subjected to interfacial Dzyaloshinskii-Moriya interaction (IDMI) are considered. It was found that three-magnon interaction coefficients in the ``Damon-Eshbach'' geometry are nonzero only in the case of the noncollinear interacting spin waves, and vanish in the case of the collinear spin waves. It was also found that the nonlinear spin wave frequency shift caused by the four-magnon interaction is nonreciprocal, and has the sign opposite to that of the nonreciprocal term in linear spin wave dispersion, so that the IDMI-induced nonreciprocity of the spin wave spectrum decreases with the increase of the spin wave amplitude.