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

This is the authors' abstract. We don't add key points for this paper.

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.