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Floquet Hofstadter butterfly in trilayer graphene with a twisted top layer

The magnetic field generated Hofstadter butterfly in single-twist trilayer graphene (TLG) is investigated using circularly polarized light (CPL) and longitudinal light emanating from a waveguide. We show that single-twist TLG has two distinct chiral limits in the equilibrium state, and the central branch of the butterfly splits into two precisely degenerate components. The Hofstadter butterfly appears to be more discernible. We also discovered that CPL causes a large gap opening at the central branch of the Hofstadter butterfly energy spectrum and between the Landau levels, with a clear asymmetry corresponding to energy $E = 0$. We point out that for right-handed CPL, the central band shifts downward, in stark contrast to left-handed CPL, where the central band shifts upward. Finally, we investigated the effect of longitudinally polarized light, which originates from a waveguide. Interestingly, we observed that the chiral symmetries of the Hofstadter butterfly energy spectrum are broken for small driving strengths and get restored at large ones, contrary to what was observed in twisted bilayer graphene.