Quantum-Geometric Design of Lattice Generalized Landau Levels

We design lattice models with tailored quantum geometry, including generalized Landau levels (LLs) satisfying the integrated trace condition and higher-Chern bands with ideal quantum geometry. Our models with $N=2$, $3$, and $4$ sublattices include a generalized Haldane model ($N=2$ honeycomb lattice model) with Gaussian-decaying hoppings realizable in twisted bilayer MoTe$_2$, and $N \geq 3$ models with exponentially decaying hoppings. Exact diagonalization reveals fractional Chern insulators in the generalized zeroth LL bands of all three models, a Moore-Read state in the generalized first LL band of the $N=4$ model, and various interaction-driven topological phases$\unicode{x2013}$including integer and fractional anomalous Hall crystals and a multicomponent Halperin state$\unicode{x2013}$in the ideal higher-Chern band of the $N=3$ model. Informed by quantum geometry, our work provides a pathway for lattice realizations of Landau-level and beyond-Landau-level physics.