  Existing Quantum Monte Carlo studies have investigated the properties of
fermions on a Lieb (CuO$_2$) lattice interacting with an on-site, or
near-neighbor electron-electron coupling. Attention has focused on the
interplay of such interactions with the macroscopic degeneracy of local zero
energy modes, from which Bloch states can be formed to produce a flat band in
which energy is independent of momentum. The resulting high density of states,
in combination with the Stoner criterion, suggests that there should be
pronounced instabilities to ordered phases. Indeed, a theorem by Lieb
rigorously establishes the existence of ferrimagnetic order. Here we study the
charge density wave phases induced by electron-phonon coupling on the Lieb
lattice, as opposed to previous work on electron-electron interactions. Our key
result is the demonstration of charge density wave (CDW) phases at one-third
and two-thirds fillings, characterized by long-range density density
correlations between doubly occupied sites on the minority or majority
sublattice, and an accompanying gap. We also compute the transition temperature
to the ordered phase as a function of the electron-phonon coupling.
