Recent observations of the fractional anomalous quantum Hall effect in moiré materials have reignited the interest in fractional Chern insulators (FCIs). The chiral limit in which analytic Landau-level-like single-particle states form an “ideal” Chern band and local interactions lead to Laughlin-like FCIs at 1/3 filling has been very useful for understanding these systems by relating them to the lowest Landau level. We show, however, that, even in the idealized chiral limit, a fluctuating quantum geometry is associated with strongly broken symmetries and a phenomenology very different from that of Landau levels. In particular, particle-hole symmetry is strongly violated and, e.g., at 2/3 filling an emergent interaction driven Fermi liquid state with no Landau level counterpart is energetically favored. In fact, even the exact Laughlin-like zero modes at 1/3 filling have a nonuniform density tracking the underlying quantum geometry. Switching to a Coulomb interaction, the ideal Chern band with electron filling of 1/4 features trivial charge density wave states. Moreover, applying a particle-hole transformation reveals that the ideal Chern band with hole filling of 3/4 supports a quantum anomalous Hall crystal with quantized Hall conductance of 𝑒2/ℎ. These phenomena have no direct lowest Landau level counterpart.
