We theoretically investigate the influence of lattice distortion effects on possible topological phases in (LaNiO${}_3$)${}_2$/(LaAlO${}_3$)${}_N$ heterostructures grown along the [111] direction. At the Hartree-Fock level, topological phases originate from an interaction-generated effective spin-orbit coupling that opens a gap in the band structure. For the unstrained system, there is a quadratic band touching at the $\Gamma$ point at the Fermi energy for spin-unpolarized electrons and Dirac points at K, K${}^{\prime }$ at the Fermi energy for fully spin-polarized electrons. Using density functional theory we first show that the quadratic band touching and Dirac points are remarkably stable to internal strain-induced out-of-plane distortions and rotations of the oxygen octahedra, which we compute. The lack of a gap opening implies that the mean-field predictions for topological phases for both the polarized and unpolarized systems are little affected by internal strain and lattice relaxations. On the other hand, we also discuss two types of lattice distortions, which have an important effect on the electronic structure. First, an external strain imposed along the [001] cubic direction can open a gap at the $\Gamma$ point, thereby stabilizing a trivial insulating phase in the spin-unpolarized system. However, it leaves the Dirac points intact. As a result, the Hartree-Fock calculation for an effective model using parameters relevant to LaNiO${}_3$ finds that symmetry-breaking strain favors a phase with polarized orbitals and antiferromagnetic spin order, but leaves earlier predictions for a zero-magnetic field topological quantum Hall state essentially unchanged. Second, we identify a possible breathing distortion of the oxygen cages stabilized by correlation effects. Such a distortion opens a gap at the Dirac points and we demonstrate that it would compete with the topological phase in the fully spin-polarized system. Taken together, our results suggest that distortion effects in thin films grown along the [111] direction in perovskites have rather different consequences as compared to those grown along [001].

• Received 20 June 2013

DOI:https://doi.org/10.1103/PhysRevB.88.115146