Fractal Spectrum in Twisted Bilayer Optical Lattice

We investigate the full spectrum of twisted bilayer optical lattices (TBOLs) across all possible twist angles. Our calculation departs from the conventional moiré physics paradigm, which focuses on continuum theory at a fixed small twist angle. We discover that the full spectrum of a TBOL exhibits an astonishing fractal structure that resembles Hofstadter's butterfly but with vanishing first Chern number. Yet unlike conventional butterfly band structures, these fractal bands emerge purely from the geometric moiré effects and do not require any external magnetic field. By mapping TBOL to a generalized Hofstadter model with long-range hopping, we reveal a universal algebraic structure underlying both systems. We also provide numerical evidence on the infinite recursive structures of the spectrum and present the algorithm for computing these structures.