Quasiperiodic hopping on the honeycomb lattice

Applying on-site quasiperiodic potentials to ultracold atomic gases can induce localization phenomena in a deterministic way, and provides a route to emulate moire materials. In this work, we introduce a quasiperiodic tunneling model on the honeycomb optical lattice, which we refer to as "quasiperiodic hopping", and demonstrate its magic-angle behavior and diffusive dynamical properties in the weakly interacting ultracold Bose gas. We construct a tight-binding model, and treat interactions on the mean field level by solving the Gross-Pitaevskii equation. In the non-interacting limit, we compute the density of states to reveal a reentrant semimetallic phase with an intermediate metallic phase transition at a critical hopping strength. This transition is consistent with a magic-angle transition in the band structure, which is analogous to generating flat bands in twisted bilayer graphene. Evidence of this transition is also shown using the structure of the eigenstates, which showcase an inversion of the positive and negative semimetal modes after the intermediate metallic transition. In the interacting case, dynamics of wavepackets are presented as the model goes through the magic-angle transition. These results establish quasiperiodic hopping as a new way to study moire-like physics in ultracold atoms on the honeycomb lattice.