Flatband and Kähler Geometry: Moiré Fractionalization, Geometric Response and Lattice Realizations

The fractional quantum Hall effect is one of the few strongly correlated quantum phases where the nature of its correlations is relatively well understood, thanks to the elegant geometric structure of Landau levels such as holomorphicity and non- commutativity. Extending this understanding to other realms of correlated quantum phases presents an open but exciting challenge. This talk reviews recent progress in bridging quantum geometry, differential geometry, and correlated matter. We begin with an overview of the recently introduced notions of the Kähler band, ideal geometry, and vortexability, which aim to identify geometric conditions that enable Bloch bands to closely resemble the lowest Landau levels. We also systematically construct their higher Landau level counterparts, termed generalized Landau levels. The implications of generalized Landau levels for the experimentally discovered fractional Chern insulators in twisted MoTe2 and pentalayer graphene materials are explored. Additionally, we examine the geometric response of generalized Landau levels, with a particular focus on Hall viscosity, a curvature of moduli space. We present an exact derivation of Hall viscosity and discuss its universality. Finally, we introduce lattice models that realize Landau levels of arbitrary Landau level index, drew from sum rules pioneered by Perelomov, and discuss its potential experimental realization and implication to correlated phases on lattices.