Spin waves (SWs) have became the subject of an intense theoretical and experimental investigation due to their potentiality as dissipationless information carriers for spintronic logic gates and waveguides. Differently from the Fourier analysis of a system’s magnetic response after proper excitation, the Hamiltonian approach  allows the computation of the whole set of SW modes, independently of the excitation symmetry and action, as an eigenvalue/eigenvector problem; moreover, the modes can be in principle computed arbitrarily close to the critical field for any magnetization change (“transition”), e.g. magnetization reversal, vortex-to saturation transition, etc. The last property is particularly suitable to the calculation of soft modes , i.e. SWs with a frequency going to zero at the critical field: at the critical field, this modes are known to trigger the transition by transferring their symmetry to the static magnetization, determining a specific instability that leads the system to reconfigure in a different way. Besides the theoretical interest in describing many kind of changes of the magnetization configuration, soft modes have surprising properties of great importance for spintronics, as a asymmetric broadening of their bandwidth  (with different group velocity in different directions), and for a dynamic explanation of the complexity of reversal avalanches (Dirac strings) in macrospin networks like artificial quasicrystals and artificial spin ices .  L. Giovannini, F. Montoncello and F. Nizzoli, Phys. Rev B 75, 024416 (2007)  F. Montoncello et al., Phys. Rev. B 77, 214402 (2008)  F. Montoncello and L. Giovannini, Appl. Phys. Lett. 104, 242407 (2014)  F. Montoncello et al., Journal of Magnetism and Magnetic Materials 423, 158 (2017).
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