| Summary: | A new class of 2d materials, moiré superlattices, has emerged and become one of the most exciting playgrounds for the study of many-body physics. These systems, thanks to their unprecedented tunability, have exhibited a plethora of interesting phenomena in experiments, such as unconventional superconductivity, strange metal behavior, exciton condensation, emergent Kondo lattice physics, quantum Hall ferromagnetism, (fractional) quantum anomalous Hall effect, and formation of anomalous Hall crystals. Typically in many of these systems, there is a nearly flat low-energy band with non-zero Berry curvature. This generalizes the familiar quantum Hall physics to a broader context, where both dispersion and quantum geometry can be varied. This thesis focuses on novel quantum phases in systems where three ingredients, kinetic energy, interaction, and band topology all play a role. First, we demonstrate quantum Hall ferromagnetism in a topological band, which is a simple yet striking example of the crucial role of band topology in the consequence of strong correlations. Motivated by this, we present a fruitful framework to think about the effects of interaction within a topologically nontrivial band, known as non-commutative field theory. This development provides an analytical handle to this broad class of challenging problems and settles long-standing puzzles shrouding quantum Hall physics. Most of the existing studies focus on the strong correlation effect on a stage defined by band topology. However, this picture is only justified when the single-particle band gap dominates over interaction. Going beyond this regime, we study two representative moire systems (1) Quantum anomalous Hall effect in transition metal dichalcogenide moire and (2) Fractional quantum anomalous Hall effect in multilayer rhombohedral graphene moire. In these systems, instead of playing a role on the stage defined by the band topology, the interaction is strong enough to determine the band topology. We identify various mechanisms for interactions to stabilize a Chern band.
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