We study ultracold fermions in an optical lattice in two different settings: with cavity-mediated long-range interactions and with time-periodic driving.

In the cavity-coupled system we consider fermions with two hyperfine states and investigate the interplay of the superradiant checkerboard density-wave phase with Fermi-liquid and Mott-insulator phases, including a region of coexistence.

On the other hand, we investigate a two-dimensional Floquet system that has been realized in the experiment of Wintersperger et al., Nat. Phys. 16, 1058 (2020), and study the impact of disorder on its topology. We find an emerging Anomalous Floquet Anderson Insulator (AFAI) phase, where all bulk states are localized, while the pumped charge in a Laughlin pump setup remains quantized.