Fractal spectrum in twisted bilayer optical lattice

The translation symmetry of a lattice is greatly modified when subjected to a perpendicular magnetic field [Zak, Phys. Rev. \textbf{134}, A1602 (1964)]. This change in symmetry can lead to magnetic unit cells that are substantially larger than the original ones. Similarly, the translation properties of a double-layered lattice alters drastically while two monolayers are relatively twisted by a small angle, resulting in large-scale moir\'{e} unit cells. Intrigued by the resemblance, we calculate the complete band structures of a twisted bilayer optical lattice and show that the geometric moir\'{e} effect can induce fractal band structures. The fractals are controlled by the twist angle between two monolayers and are closely connected to the celebrated butterfly spectrum of two-dimensional Bloch electrons in a magnetic field [Hofstadter, Phys. Rev. B \textbf{14}, 2239 (1976)]. We demonstrate this by proving that the twisted bilayer optical lattice can be mapped to a generalized Hofstadter's model with long-range hopping. Furthermore, we provide numerical evidence on the infinite recursive structures of the spectrum and give an algorithm for computing these structures.