Charge Transport in the Non-ergodic Extended phase of Quasi-periodic systems

We study the transport properties and the spectral statistics of a one-dimensional closed quantum system of spinless fermions in a quasi-periodic potential which produces a single particle mobility edge. For such systems, it has been shown that many body eigenstates can be of three different kinds: extended and ETH (energy thermalization hypothesis) obeying (thermal), localized and ETH violating (many body localized) and extended and ETH violating (non-ergodic extended). Here we investigate the non-ergodic extended phase from the point of view of level spacing statistics and charge transport. We calculate the DC conductivity and the low frequency conductivity σ(ω) and show that both are consistent with sub-diffusive transport. This is contrasted with diffusive transport in the thermal phase and blocked transport in the MBL phase.