  Density functional theory calculations with an on-site Coulomb repulsion term
(GGA+U method) reveal competing ground states in (111) oriented
(LaAlO$_3$)$_M$/(SrTiO$_3$)$_N$ superlattices with n-type interfaces, ranging
from spin, orbital polarized, Dirac point Fermi surface to charge ordered flat
band phases. These are steered by the interplay of (i) Hubbard U, (ii)
SrTiO$_3$ quantum well thickness and (iii) crystal field spitting tied to
in-plane strain. In the honeycomb lattice bilayer case N=2 under tensile strain
inversion symmetry breaking drives the system from a ferromagnetic Dirac point
(massless Weyl semimetal) to a charge ordered multiferroic (ferromagnetic and
ferroelectric) flat band massive (insulating) phase. With increasing SrTiO$_3$
quantum well thickness an insulator-to-metal transition occurs.
