概要
The rise of moir? materials has led to experimental realizations of integer and fractional Chern insulators in small[1] or vanishing magnetic fields[2]. At the same time, a set of minimal conditions sufficient to guarantee an Abelian fractional state in a flat band was identified, namely "ideal" or "vortexable" quantum geometry[3,4]. Such vortexable bands share essential features with the lowest Landau level, 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 (1LL), where non-Abelian states at ν = 1/2, 2/5 are known to be competitive[5]. 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[6], 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. We apply the definition to identify two models with 1LL structure ? a toy model of double bilayer twisted graphene and a more realistic model of strained Bernal graphene.