Flux-Switching Floquet Engineering

We study a square-lattice Harper–Hofstadter model driven by periodic switching of the magnetic flux at arbitrary drive period. In our protocol the flux per plaquette alternates among rational values {p_j/q_j}, yielding a Floquet quasienergy spectrum with Q bands, where Q is the least common multiple of the denominators of the rational flux values {q_j} . For the π ⁣→ ⁣−π switching routine with next-nearest-neighbor hopping, we obtain closed-form quasienergies and band Chern numbers. We further compute the gap winding invariants W₀​ and W_π and identify a window, beyond a critical next-nearest-neighbor amplitude, that hosts robust anomalous Floquet edge modes. We visualize hybrid Floquet–Hofstadter butterflies with gaps labeled by their topological invariants, and we adapt a Floquet–Středa response that links density to multi-flux driving. Together, these results provide quantitative predictions for multi-flux Floquet drives, with direct relevance to synthetic-gauge platforms such as cold-atom systems.