The magnetorotational instability (MRI) is the most promising mechanism by which angular momentum is efficiently transported outwards in astrophysical discs. However, its application to protoplanetary discs remains problematic. These discs are so poorly ionised that they may not support magnetorotational turbulence in regions referred to as `dead zones. It has recently been suggested that the Hall effect, a non-ideal magnetohydrodynamic (MHD) effect, could revive these dead zones by enhancing the magnetically active column density by an order of magnitude or more. We investigate this idea by performing local, three-dimensional, resistive Hall-MHD simulations of the MRI in situations where the Hall effect dominates over Ohmic dissipation. As expected from linear stability analysis, we find an exponentially growing instability in regimes otherwise linearly stable in resistive MHD. However, instead of vigorous and sustained magnetorotational turbulence, we find that the MRI saturates by producing large-scale, long-lived, axisymmetric structures in the magnetic and velocity fields. We refer to these structures as zonal fields and zonal flows, respectively. Their emergence causes a steep reduction in turbulent transport by at least two orders of magnitude from extrapolations based upon resistive MHD, a result that calls into question contemporary models of layered accretion. We construct a rigorous mean-field theory to explain this new behaviour and to predict when it should occur. Implications for protoplanetary disc structure and evolution, as well as for theories of planet formation, are briefly discussed.
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