Flux-switching Floquet engineering

We present an analysis of a square-lattice Harper-Hofstadter model with a periodically varying magnetic flux with time. By switching the dimensionless flux per plaquette between a set of values $\{p_j/q_j\}$ the Floquet quasienergy spectrum is folded into Q = lcm$\{q_j\}$ bands. We determine closed form analytical solutions for the quasienergy spectrum and Chern numbers for the -1/2 $\to$ 1/2 flux switching case, as well as the Rudner-Lindner-Berg-Levin (RLBL) winding invariants W numerically, and construct the corresponding topological phase diagram for arbitrary driving period. We find that generic flux-switching drives feature interlaced Hofstadter butterfly quasienergy spectra, and the gaps in the spectrum may be labeled according to a Diophantine equation which relates the quasienergy gap index to the fluxes attained in the drive and their associated per-step windings.