# Achievement

## Experimental realization of Hofstadters Butterfly

### Project

IGERT: Engineering Photons for a Sustainable Future

### University

Columbia University

(New York, NY)

### PI

## Trainee Achievements

### Experimental realization of Hofstadters Butterfly

Electrons moving in a periodic electric potential form Bloch energy bands where the mass of electrons are effectively changed. In a strong magnetic field, the cyclotron orbits of free electrons are quantized and Landau levels forms with a massive degeneracy. These two quantization effects can lead into a self-similar fractal set of energy spectrum known as “Hofstadter’s Butterfly.” The recent advent of graphene, where its Bloch electrons can be described by Dirac feremions, provides a new opportunity to investigate this half century old problem experimentally. The IGERT trainee has demonstrated the experimental realization of Hofstadter’s Butterfly via substrate engineered graphene under extremely high magnetic fields controlling two competing length scales governing Dirac-Bloch states and Landau orbits, respectively. Here, the strong Coulomb interactions and approximate spin-pseudo spin symmetry are predicted on variety of integer quantum Hall ferromagnetic by the IGERT trainee.

- “Trainee Achievements”
- Achievements for this Project