Dynamic off-centering of Cr$^{3+}$ ions and short-range magneto-electric clusters in in CdCr$_2$S$_4$

The cubic spinel CdCr$_2$S$_4$ gained recently a vivid interest, given the relevance of relaxor-like dielectric behavior in its paramagnetic phase. By a singular combination of local probe techniques namely Pair Distribution Function and Perturbed Angular Correlation we firmly establish that the Cr ion plays the central key role on this exotic phenomenon, namely through a dynamic off-centering displacement of its coordination sphere. We further show that this off centering of the magnetic Cr-ion gives rise to a peculiar entanglement between the polar and magnetic degrees of freedom, stabilizing, in the paramagnetic phase, short range magnetic clusters, clearly seen in ultra-low field susceptibility measurements. Moreover, the Landau theory is here used to demonstrate that a linear coupling between the magnetic and polar order parameters is sufficient to justify the appearance of magnetic cluster in paramagnetic phase of this compound. These results open insights on the hotly debated magnetic and polar interaction, setting a step forward in the reinterpretation of the coupling of different physical degrees of freedom.