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Relativistic dynamical spin excitations of magnetic adatoms

We present a first-principles theory of dynamical spin excitations in the presence of spin-orbit coupling. The broken global spin rotational invariance leads to a new sum rule. We explore the competition between the magnetic anisotropy energy and the external magnetic field, as well as the role of electron-hole excitations, through calculations for 3$d$-metal adatoms on the Cu(111) surface. The spin excitation resonance energy and lifetime display non-trivial behavior, establishing the strong impact of relativistic effects. We legitimate the use of the Landau-Lifshitz-Gilbert equation down to the atomic limit, but with parameters that differ from a stationary theory.