The recent discovery of magnetic topological insulators has opened new avenues to explore exotic states of matter that can emerge from the interplay between topological electronic states and magnetic degrees of freedom, be it ordered or strongly fluctuating. Motivated by the effects that the dynamics of the magnetic moments can have on the topological surface states, we investigate the magnetic fluctuations across the (MnBi$_{text{2}}$Te$_{text{4}}$)(Bi$_{text{2}}$Te$_{text{3}}$)$_{text{n}}$ family. Our paramagnetic electron spin resonance experiments reveal contrasting Mn spin dynamics in different compounds, which manifests in a strongly anisotropic Mn spin relaxation in MnBi$_{text{2}}$Te$_{text{4}}$ while being almost isotropic in MnBi$_{text{4}}$Te$_{text{7}}$. Our density-functional calculations explain these striking observations in terms of the sensitivity of the local electronic structure to the Mn spin-orientation, and indicate that the anisotropy of the magnetic fluctuations can be controlled by the carrier density, which may directly affect the electronic topological surface states.
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