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We discuss the results of $^{75}$As Nuclear Quadrupole Resonance (NQR) andmuon spin relaxation measurements in AFe$_2$As$_2$ (A= Cs, Rb) iron-basedsuperconductors. We demonstrate that the crossover detected in the nuclearspin-lattice relaxation rate $1/T_1$ (around 150 K in RbFe$_2$As$_2$ and around75 K in CsFe$_2$As$_2$), from a high temperature nearly localized to a lowtemperature delocalized behaviour, is associated with the onset of aninhomogeneous local charge distribution causing the broadening or even thesplitting of the NQR spectra as well as an increase in the muon spin relaxationrate. We argue that this crossover, occurring at temperatures well above thephase transition to the nematic long-range order, is associated with a chargedisproportionation at the Fe sites induced by competing Hund's and Coulombcouplings. In RbFe$_2$As$_2$ around 35 K, far below that crossover temperature,we observe a peak in the NQR $1/T_1$ which is possibly associated with thecritical slowing down of electronic nematic fluctuations on approaching thetransition to the nematic long-range order.

Charge and nematic orders in AFe$_2$As$_2$ (A= Rb,Cs) superconductors