Fractal defect states in the Hofstadter butterfly

TL;DR

This paper explores how introducing vacancies in a square lattice under a magnetic field creates defect states within the fractal Hofstadter butterfly energy gaps, revealing their localization and magnetic properties.

Contribution

It demonstrates that vacancies induce defect energy levels across all fractal gaps and characterizes their localization and magnetic moments, linking defect states to the fractal energy structure.

Findings

Vacancies create defect levels in all Hofstadter gaps.

Defect wavefunctions exhibit fractal scaling of localization lengths.

Each defect state has a unique orbital magnetic moment.

Abstract

We investigate the electronic properties in the Bloch electron on a square lattice with vacancies in the uniform magnetic field. We show that a single vacancy site introduced to the system creates a defect energy level in every single innumerable fractal energy gap in the Hofstadter butterfly. The wavefunctions of different defect levels have all different localization lengths depending on their fractal generations, and they can be described by a single universal function after an appropriate fractal scaling. We also show that each defect state has its own characteristic orbital magnetic moment, which is exactly correlated to the gradient of the energy level in the Hofstadter diagram. Probing the spatial nature of the defect-localized states provides a powerful way to elucidate the fractal nature of the Hofstadter butterfly.