Mobility-edge-embedded Hofstadter butterfly from a tilt-induced quasiperiodic potential

The Hofstadter butterfly (HB) and mobility edges (MEs) are hallmark phenomena of quasiperiodic systems, yet their interplay remains elusive. Here, we demonstrate their coexistence within a tilt-induced quasiperiodic potential on a square lattice, giving rise to a ``mobility-edge-embedded Hofstadter butterfly'' (MEE-HB). This potential is generated by aligning a periodic potential at an angle relative to the lattice axes -- a configuration readily accessible in optical lattice experiments. Using a tight-binding model, we show that the MEE-HB manifests as a fractal energy splitting pattern hosting MEs that separate extended and localized states. Our Harper-like equation shows that the fractal pattern originates from one-dimensional quasiperiodic potentials, while MEs stem from effective long-range hopping. Notably, the MEE-HB exhibits a fractal dimension of $0.8$--$1.0$, significantly exceeding the $0.4$--$0.6$ range of the standard butterfly, indicating a denser spectrum. Our findings establish tilt-induced potentials as a versatile platform for exploring the interplay between fractal structures and localization.