Exact Lattice Construction of All Landau Level States

The Kapit-Mueller model is a variant of the Hofstadter problem with designed tight-binding couplings that allows exact flat band corresponds to lowest Landau level states on lattice sites. The derivation of the Kapit-Mueller model relies on Poisson summation, a lattice sum rule for general holomorphic functions, initially pointed out by Perelomov. In this work, we generalize the Perelomov identity and expand the Kapit-Mueller construction beyond lowest Landau level by introducing a class of Hamiltonian whose exact flat band states are Landau level states of arbitrary Landau level index. This proposed analytical model simultaneously exhibits lattice and Landau level characteristics. By including interactions, it has the potential to realize non-Abelian states in cold atom settings and offers a promising bridge between fractional quantum Hall and quantum spin liquid physics in the long term.