Persistent non-ergodic fluctuations in mesoscopic insulators: The NSS model in the unitary and symplectic ensembles

We give a detailed picture of the mesoscopic conductance fluctuations in the deep insulating regime (DIR) within the Nguyen, Spivak and Shklovskii model in the unitary and symplectic ensembles. Slutski's theorem is invoked to rigorously state the ergodic problem for conductance fluctuations in the DIR, in contrast with previous studies. A weakly decaying behavior of the log-conductance correlation function, even weaker when spin-orbit scatterers are included, is established on the relevant field scale of the model. Such a slow decay implies that the stochastic process, defined by the fluctuations of the log-conductance, is non-ergodic in the mean square sense in the ensembles with the reported symmetries. The results can be interpreted in terms of the effective number of samples within the available magnetic scale. Using the replica approach, we derive the strong localisation counterparts of the well known 'cooperon' and 'diffuson' which permit analyzing quantitatively the decaying behavior of the correlation function and reveal its symmetry related properties in agreement with the numerical results.