The intermediate coupling regime of the quantum many-body problem – where kinetic and potential energy compete on the same order – is notoriously difficult. Analytical approaches there are generically uncontrolled, and numerical tools are often insufficient. However, experiments on the growing array of two-dimensional materials routinely probe this regime. To understand such experiments, a new intermediate coupling toolkit must be developed.
This talk will focus on recent experiments in magic-angle graphene that detected fractional Chern insulators. Fractional Chern insulators (FCIs) realize the remarkable physics of the fractional quantum Hall effect in crystalline systems with Chern bands, casting off the typical requirements of Landau levels and strong magnetic fields. Understanding these intermediate-coupling experiments required an "all of the above" approach, combining close experimental collaborations with new theoretical and numerical tools. I will start by briefly overviewing the experimental phenomenology: in small magnetic fields, FCIs compete with charge-density waves due to the significant band dispersion. Next I will introduce ``vortexable bands", a class of beyond-Landau level systems with exact FCI ground states. Magic-angle graphene is vortexable in a limit, making the realistic system almost ideal for FCIs. Finally, I will use the new numerical technique of ``MPO compression" to compute the phase diagram of FCIs in magic-angle graphene, predicting where experiments should look to find FCIs in zero external magnetic field.