Electron-phonon cooling power in Anderson insulators

First microscopic theory for electron-phonon energy exchange in Anderson insulators is developed. The major contribution to the cooling power as a function of electron temperature is shown to be directly related to the correlation function of the local density of electron states at small energy difference argument. In Anderson insulators not far from localization transition, this correlation function is strongly enhanced by wave-function's multi-fractality and, additionally, by the presence of Mott's resonant pairs of localized states. The theory we develop explains huge enhancement of the cooling power observed in insulating Indium Oxide films as compared to predictions of the theory previously developed for disordered metals. Our results open the way to predict the conditions appropriate for the observation of Many Body Localization transition those presence in electronic insulators was advocated in the seminal paper by Basko, Aleiner and Altshuler (2006) but have not been convincingly demonstrated yet.