Higher Vortexability: Zero-Field Realization of Higher Landau Levels

The rise of moiré materials has led to experimental realizations of integer and fractional Chern insulators in small or vanishing magnetic fields. At the same time, a set of minimal conditions sufficient to guarantee an Abelian fractional state in a flat band were identified, namely, "ideal" or "vortexable" quantum geometry. Such vortexable bands share essential features with the lowest Landau level (LL), while excluding the need for more fine-tuned aspects such as flat Berry curvature. A natural and important generalization is to ask if such conditions can be extended to capture the quantum geometry of higher Landau levels, particularly the first LL (1LL), where non-Abelian states at ν=1/2,2/5 are known to be competitive. The possibility of realizing these states at zero magnetic field, and perhaps even more exotic ones, could become a reality if we could identify the essential structure of the 1LL in Chern bands. In this work, we introduce a precise definition of 1LL quantum geometry, along with a figure of merit that measures how closely a given band approaches the 1LL. Periodically strained Bernal graphene is shown to realize such a 1LL structure even in zero magnetic field.