🦋 Scientists observe 'Hofstadter's butterfly' for the first time

• Scientists from Princeton University have measured the energy levels of electrons in a new quantum material and discovered that they follow a fractal pattern.
• The quantum version of the fractal pattern, known as "Hofstadter's butterfly," has been theoretically predicted since 1976 but has now been directly observed for the first time in a real material.
• The discovery was made by accident when researchers were investigating superconductivity in twisted graphene layers.

What is Hofstadter's butterfly?

Hofstadter's butterfly represents a fractal pattern that was predicted in a pioneering study from 1976 by Douglas Hofstadter. He predicted that the energy levels of electrons in two-dimensional crystals under the influence of a strong magnetic field would display a characteristic fractal energy spectrum. The name "butterfly" is used because the pattern, when plotted against energy and magnetic field, shows an elegant and intricate configuration resembling a butterfly's wings.

Fractals are self-repeating patterns that occur on different length scales and can be seen in nature in various settings, including snowflakes, ferns, and coastlines. However, fractals are uncommon in the quantum world.

The discovery happened by accident

The research team, led by Professor Ali Yazdani, made the discovery while actually investigating superconductivity in twisted bilayer graphene.

The researchers were trying to study superconductivity, but when creating their samples, they got a moiré pattern with a longer periodicity than intended. This mistake turned out to be exactly what was needed to observe Hofstadter's spectrum.

New technology made the observation possible

The study was made possible thanks to a new method in materials science, which involved stacking and twisting two layers of carbon atoms to create an electron pattern resembling a French textile design known as moiré.

The team used a scanning tunneling microscope to image moiré crystals at atomic resolution and examine their electron energies. The microscope works by bringing a sharp metal tip less than a nanometer from the surface to allow quantum "tunneling" of electrons from the tip to the sample.

This microscope was particularly crucial because the tool is especially sensitive to the energy of electrons in materials. Myungchul Oh, a postdoctoral researcher and co-author of the study, noted: "The scanning tunneling microscope is a direct energy probe, which helps us relate back to Hofstadter's original calculations of energy levels. Previous studies on Hofstadter's butterfly were based on electrical resistance measurements that don't measure energy."

Significance for fundamental physics

While the research may not yield any practical applications immediately, the work revealed features of Hofstadter's spectrum that are of interest to fundamental physics research. The researchers found that theoretical modeling of the spectrum improved if they included phenomena related to electrons interacting with each other, an important aspect that was left out of Hofstadter's original calculations.

Including the impact of such interactions is difficult, and experiments become particularly valuable in understanding the many-electron version of this problem. The experimental team worked in close collaboration with a theoretical team led by Professor Biao Lian from the physics department and his students, who are also co-authors of the paper.

Michael Scheer, a graduate student in physics at Princeton and one of the paper's lead authors, noted: "The Hofstadter regime is a rich spectrum of topological states, and I think being able to image these states can be a very powerful way to understand their quantum properties."

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