Floquet states in (LaNiO$_3$)$_2$/(LaAlO$_3$)$_N$ heterostructures grown along the (111) direction
Using Floquet-Bloch theory we study the effect of circularly and linearly polarized light on the electronic structure of (LaNiO$_3$)$_2$/(LaAlO$_3$)$_N$ heterostructure grown along the (111) direction. In equilibrium, a tight-binding fit to the first principles band structure shows that nearest-neighbor hopping plays a dominant role while second-neighbor hopping breaks the particle-hole symmetry and determines the finer band features. The four bands of the LaNiO$_3$ bilayer exhibit both quadratic band touching points and Dirac points. By varying the amplitude of the incident light, one can independently tune the first and second-neighbor hopping for fixed frequency, which leads to considerable control over the Floquet band structure. We investigate this control in detail, and study how the quadratic and Dirac band touchings are influenced by the polarization and intensity of the light. We derive effective 2-band Hamiltonians (for both quadratic and Dirac band touching points) that accurately captures these results. We further study an extended model which explicitly includes oxygen $p$-orbitals and compare the results to the effective model that contains only the nickel $d$-orbitals. We conclude with a computation of the frequency dependent optical Hall conductivity using the full four band model and analyze the various inter-band contributions of the Floquet modes.