  We study spinless fermions with nearest-neighbor repulsive interactions
($t$-$V$ model) on the two-dimensional three-band Lieb lattice. At
half-filling, the free electronic band structure consists of a flat band at
zero energy and a single cone with linear dispersion. The flat band is expected
to be unstable upon inclusion of electronic correlations, and a natural channel
is charge order. However, due to the three-orbital unit cell, commensurate
charge order implies an imbalance of electron and hole densities and therefore
doping away from half-filling. Our numerical results show that below a
finite-temperature Ising transition a charge density wave with one electron and
two holes per unit cell and its partner under particle-hole transformation are
spontaneously generated. Our calculations are based on recent advances in
auxiliary-field and continuous-time quantum Monte Carlo simulations that allow
sign-free simulations of spinless fermions at half-filling. It is argued that
particle-hole symmetry breaking provides a route to access levels of finite
doping, without introducing a sign problem.
