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

Spontaneous particle-hole symmetry breaking of correlated fermions on the Lieb lattice