Effect of band filling and symmetry breaking on the electronic ground state in (La$X$O$_{3})_{2}$/(LaAlO$_{3})_{4}$(111) ($X=$3d) superlattices*

Structural patterns, e.g. a buckled honeycomb lattice, realized in (111)-oriented perovskite bilayers may lead to exotic electronic ground states such as a Dirac-point Fermi surface [1-3]. Based on density functional theory calculations including a Hubbard $U$ term, we explore systematically the effect of band filling in [111]-oriented (La$X$O$_{3})_{2}$/(LaAlO$_{3})_{4}$ superlattices with $X$ spanning the series of open shell \textit{3d} ions. The interplay of charge, spin, orbital, and lattice degrees of freedom reveals some regularities over the series, but also several unexpected symmetry lowering reconstructions that can guide the design of artificial materials of desired spin-charge-orbital order in conjunction with size of the energy gap and the possibility for emergent topological character. \\[4pt] [1] D. Xiao, et al., Nat. Commun. \textbf{2}, 596 (2011).\\[0pt] [2] D. Doennig, W. E. Pickett, and R. Pentcheva, Phys. Rev. Lett. \textbf{111}, 126804 (2013).\\[0pt] [3] D. Doennig, W. E. Pickett, and R. Pentcheva, Phys. Rev. B \textbf{89}, 121110(R) (2014).