A team of scientists from Princeton University has measured the energies of electrons in a new class of quantum materials and has found them to follow a fractal pattern. Fractals are self-repeating patterns that occur on different length scales and can be seen in nature in a variety of settings, including snowflakes, ferns, and coastlines. The image shows a quantum version of a fractal pattern, known as “Hofstadter’s butterfly,” which has long been predicted, but the new study marks the first time it has been directly observed experimentally in a real material. Credit: Yazdani group

Researchers Captured Elusive Quantum Hofstadter’s Butterfly Pattern

Princeton physicists accidentally discovered and directly visualized Hofstadter’s butterfly—a theoretical fractal pattern of electron energy levels predicted in 1976—using scanning tunneling microscopy on twisted bilayer graphene, revealing not only the long-sought fractal energy spectrum but also new quantum phenomena driven by electron interactions beyond Hofstadter’s original calculations.