  Inspired by the growing interest in probing many-body phases in novel
two-dimensional lattice geometries we investigate the properties of cold atoms
as they could be observed in an optical Lieb lattice. We begin by computing
Wannier functions localised at individual sites for a realistic experimental
setup, and determining coefficients for a Hubbard-like model. Based on this, we
show how experiments could probe the robustness of edge states in a Lieb
lattice with diagonal boundary conditions to the effects of interactions and
realise strongly correlated many-body phases in this geometry. We then
generalise this to interacting particles in a half-filled 1D Lieb ladder, where
excitations are dominated by flat band states. We show that for strong
attractive interactions, pair correlations are enhanced even when there is
strong mixing with the Dirac cone. These findings in 1D raise interesting
questions about the phases in the full 2D Lieb lattice which we show can be
explored in current experiments.
