Hall effect in protostellar disc formation and evolution

The Hall effect is recently shown to be efficient in magnetized dense molecular cores, and could lead to a bimodal formation of rotationally supported discs (RSDs) in the first core phase. However, how such Hall dominated systems evolve in the protostellar accretion phase remains unclear. We carry out 2D axisymmetric simulations including Hall effect and Ohmic dissipation, with realistic magnetic diffusivities computed from our equilibrium chemical network. We find that Hall effect only becomes efficient when the large population of very small grains (VSGs: $\lesssim$10 nm) is removed from the standard MRN size distribution. With such an enhanced Hall effect, however, the bimodality of disc formation does not continue into the main accretion phase. The outer part of the initial $\sim$40 AU disc formed in the anti-aligned configuration (${\bf \Omega \cdot B} 0$), disc formation is suppressed initially but a counter-rotating disc forms subsequently due to efficient azimuthal Hall drift. The counter-rotating disc first grows to $\sim$30 AU as Hall effect moves the magnetic field radially outward, but only the inner $\lesssim$10 AU RSD is long-lived like in the anti-aligned case. Besides removing VSGs, cosmic ray ionization rate should be below a few 10$^{-16}$ s$^{-1}$ for Hall effect to be efficient in disc formation. We conclude that Hall effect produces small $\lesssim$10--20 AU discs regardless of the polarity of the magnetic field, and that radially outward diffusion of magnetic fields remains crucial for disc formation and growth.