Observation of Hofstadter Butterfly a Breakthrough for Physicists

A predicted, but never-before-seen energy pattern has been observed by an international team of researchers, who confirmed the 40-year-old prediction that a butterfly-shaped energy spectrum exists in the quantum realm.

The butterfly-shaped pattern was first theorized by physicist Douglas Hofstadter in 1976, but it took the tools and technology now available at the National High Magnetic Field Laboratory (MagLab) to prove its existence

"The observation of the 'Hofstadter butterfly' marks a real landmark in condensed matter physics and high magnetic field research," said Greg Boebinger, director of the MagLab. "It opens a new experimental direction in materials research."

The Hofstadter butterfly emerges when electrons are confined to a two-dimensional sheet, and subjected to both a periodic potential energy (akin to a marble rolling on a sheet the shape of an egg carton) and a strong magnetic field. The Hofstadter butterfly is a fractal pattern-it contains shapes that repeat on smaller and smaller size scales. Fractals are common in classical systems such as fluid mechanics, but rare in the quantum mechanical world.

The Hofstadter butterfly is one of the first quantum fractals theoretically discovered in physics but, until now, there has been no direct experimental proof of this spectrum.

To confirm the Hofstadter butterfly, the research team used graphene.

Graphene is a Nobel Prize-winning material that holds tremendous promise in revolutionizing computers, batteries, cell phones, televisions and even airplanes. A one-atom thick, honeycomb array of carbon atoms, graphene is virtually see-through, yet 300 times stronger than steel and 1,000 times more conducting than silicon.

An effect called a moiré pattern arises naturally when atomically thin graphene is placed on an atomically flat boron nitride substrate, which has the same honeycomb atomic lattice structure as graphene but with a slightly longer atomic bond length.

To map the graphene energy spectrum, the team then measured the electronic conductivity of the samples at very low temperatures in extremely strong magnetic fields up to 35 Tesla (consuming 35 megawatts of power) at the MagLab. The measurements show the predicted self-similar patterns, providing the best evidence to date for the Hofstadter butterfly, and providing the first direct evidence for its fractal nature.

"Now we see that our study of moiré-patterned graphene provides a new model system to explore the role of fractal structure in quantum systems," said Cory Dean, the first author of the paper. "This is a huge leap forward-our observation that interplays between competing length scales result in emergent complexity provides the framework for a new direction in materials design. And such understanding will help us develop novel electronic devices employing quantum engineered nanostructures."

Dean said the opportunity to confirm a 40-year-old prediction in is "rare, and tremendously exciting,"

"Our confirmation of this fractal structure opens the door for new studies of the interplay between complexity at the atomic level in physical systems and the emergence of new phenomenon arising from complexity."

The research came from a collaborate effort between Columbia University, City University of New York, the University of Central Florida (UCF), and Tohoku University and the National Institute for Materials Science in Japan.

The results of the study are published in the journal Nature.