Almost 50 years ago, computer scientist Douglas Hofstadter predicted that a butterfly would spread its wings in the quantum world. Under the right conditions, tiny electrons in a quantum system could ...

In a discovery that brings a legendary quantum theory to life, physicists at Princeton University have directly observed Hofstadter’s butterfly — a fractal pattern in electron energy levels that was ...

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 ...

Random geometry and quantum gravity represent interwoven themes at the frontier of modern mathematical physics. These fields examine the stochastic nature of geometric structures that underpin quantum ...

Theoretical calculation of the allowed energy levels of electrons in twisted bilayer graphene in a magnetic field. These allowed energy levels (black dots) form a rare example of a quantum fractal ...