Broken Symmetry in Ideal Chern Bands

Recent observations of the fractional anomalous quantum Hall effect in moir\'e 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 favoured. In fact, even the exact Laughlin-like zero modes at $1/3$ filling have a non-uniform 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 $e^2/h$. These phenomena have no direct lowest Landau level counterpart.