Persistent Non-ergodic fluctuations in mesoscopic insulators

We give a detailed and rigorous picture of the mesoscopic conductance fluctuations in the deep insulating regime (DIR) within the Nguyen, Spivak and Shklovskii model including spin-orbit coupling (SO). Without SO, we find that fluctuations of the log-conductance are persistent above a saturation field Bs, where one has that the logconductance is approximately a stationary random process. In contrast, in the SO case the saturation field Bs is negligible and the stationarity is well realized. We find non-vanishing disorder fluctuations of the field average of the log-conductance as a quantitative measure of the lack of ergodicity in the mean square sense. To this fact, a weak decaying behavior of the correlation function, even weaker in the case of SO is established on the relevant field scale of the model, one flux quantum per plaquette. This finding corroborate the behavior of the fluctuations of the field average of the log-conductance and permit us to invoke Slutski’s theorem to conclude that the whole stochastic process defined by the log-conductance is non-ergodic in the mean square