Periodic and aperiodic dynamics of flat bands in diamond-octagon lattice

We drive periodically a two-dimensional diamond-octagon lattice model by switching between two Hamiltonians corresponding two different magnetic flux piercing through diamond plaquette to investigate the generation of topological flat bands. We show that in this way, the flatness and topological nature of all the bands of the model can be tuned and Floquet quasibands can be made topologically flat while its static counterpart does not support the topological band and flat band together. We define here a measure of flatness of quasienergy bands from their microscopic details that has been justified using the numerical calculations of the Floquet density of states. In the context of Floquet dynamics, we indeed have a better control on the engineering of flat bands. Interestingly, we find the generation of flux current due to periodic drives. We systematically analyze the work done and flux current in the asymptotic limit as a function of system's parameters to show that topology and flatness both share a close connection to the flux current and work done, respectively. We finally extend our investigation to the aperiodic array of step Hamiltonian, where we find that the heating up problem can be significantly reduced if the initial state is substantially flat as the initial large degeneracy of states prevents the system from absorbing energy easily from the aperiodic driving. In addition, we show that the heating can be reduced if the values of the magnetic flux in the step Hamiltonians are made small and, also the duration of these fluxes become unequal. Finally, we successfully explain our finding by plausible analytical arguments.