Fig. 1: Wave evolution in a temporal multilayer structure and two kinds of TIs.

a Illustration of the concept of a temporal multilayer structure, where a wave evolves inside a discrete system experiencing two TIs at different times. Schematic operations of band-unfolding (and band-folding as the inverse operation) and band shifting. b, c Changes of the lattice structure by changing the ranges of couplings or coupling phase for realizing operations. d, e Changes of the states and band structures accordingly. Red (violet) lines are the bandstructures before (after) the TI. Green (yellow) dots denote the states before (after) the TI.