Eigenmodes of magnetic skyrmion lattices

We explore the interplay between topology and eigenmodes by changing the stabilizing mechanism of skyrmion lattices (skX). We focus on two prototypical ultrathin films hosting an hexagonal (Pd/Fe/Ir(111)) and a square (Fe/Ir(111)) skyrmion lattice, which can both be described by an extended Heisenberg Hamiltonian. We first examine whether the Dzyaloshinkskii-Moriya, or the exchange interaction as the leading energy term affects the modes of the hexagonal skX of Pd/Fe/Ir(111). In all cases, we find that the lowest frequency modes correspond to internal degrees of freedom of individual skyrmions, and suggest a classification based on azimuthal and radial numbers $(l,p)$, with up to $l=6$, and $p=2$. We also show that the gyration behavior induced by an in-plane field corresponds to the excitation of $l=1$ deformation modes with varying radial numbers. Second, we examine the square lattice of skyrmions of Fe/Ir(111). Its stabilization mechanism is dominated by the 4-spin interaction. After relaxation, the unit cell does not carry a topological charge, and the eigenmodes do not correspond to internal skyrmion deformations. By reducing the 4-spin interaction, the integer topological charge is recovered, but the charge carriers do not possess internal degrees of freedom, nor are they separated by energy barriers. We conclude that a 4-spin dominated Hamiltonian does not yield skyrmion lattice solutions, and that therefore, a nontrivial topology does not imply the existence of skyrmions.