Realizing topological flatbands with tailored single-particle Hilbert spaces is a critical step toward exploring many-body phases, such as those featuring anyonic excitations. One prominent example is the Kapit-Mueller model, a variant of the Harper-Hofstadter model that stabilizes lattice analogs of the lowest Landau-level states. The Kapit-Mueller model is constructed based on the Poisson summation rule, an exact lattice sum rule for coherent states. In this work, we consider higher Landau-level generalizations of the Poisson summation rule, from which we derive families of parent Hamiltonians on a half-flux lattice which have exact flatbands whose flatband wave functions are lattice versions of higher Landau-level states. Focusing on generic Bravais lattices with only translation and inversion symmetries, we discuss how these symmetries enforce gaplessness and singular points for odd Landau-level series and how to achieve fully gapped parent Hamiltonians by mixing even and odd series. Our model points to a large class of tight-binding models with suitable energetic and quantum geometries that are potentially useful for realizing non-Abelian fractionalized states when interactions are included. The model exhibits rapidly decaying hopping amplitudes, making it potentially realizable with neutral atoms in optical lattices.

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• Flat bands
• Landau levels
• Mathematical physics
• Optical lattices & traps
• Quantum Hall effect
• Topological phases of matter
• Ultracold gases
• Lattice models in condensed matter
• Mathematical physics methods