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

TL;DR

This paper introduces a novel fractal energy spectrum called the mobility-edge-embedded Hofstadter butterfly (MEE-HB), arising from tilt-induced quasiperiodic potentials, revealing complex interplay between fractal patterns and localization in lattice systems.

Contribution

It demonstrates the coexistence of Hofstadter butterfly patterns and mobility edges in tilt-induced quasiperiodic potentials, a configuration accessible in optical lattice experiments, and analyzes their fractal properties.

Findings

The MEE-HB exhibits a fractal dimension of 0.8–1.0, denser than standard butterflies.

Mobility edges separate extended and localized states within the fractal spectrum.

Fractal pattern originates from one-dimensional quasiperiodic potentials and long-range hopping effects.

Abstract

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.