Almost ideal Chern bands in periodically strained graphene

We study periodically strained graphene that induces a triangular pseudo-magnetic field (PMF) with zero mean inspired by recent experiments on buckled graphene on NbSe2 substrate. We show that by combining the PMF with a matching periodic scalar field, we can realize an almost ideal isolated narrow band with valley-resolved Chern number C=+-1. By tuning the field, we show that we can achieve an exceptionally small bandwidth that is smaller by at least two orders of magnitude compared to the characteristic energy scale of the system. Furthermore, we show analytically that the band satisfies the ideal band condition up to exponentially small correction which enables us to express their wavefunctions in terms of those of the lowest Landau level in a PMF with a finite average of one flux quantum per unit cell. The combination of narrow bandwidth and ideal band geometry makes the system a perfect platform to achieve strongly correlated topological phases in a setting that is simpler and more tunable compared to conventional graphene-based moire systems. In particular, we show by means of Hartree-Fock numerics that the system realizes a quantum anomalous Hall insulator at odd integer fillings of the flat bands. Upon further fractional filling of the band, we show using exact diagonalization that the system realizes a fractional Chern insulator for parameters in the experimentally feasible range.