However, in some respects the evidence was somewhat unsatisfactory. Jitomirskaya and Avila used a method that was only applicable to certain irrational values of alpha. Combining this with the earlier indirect evidence, it could be said that the problem was solved. But this combined evidence was not elegant. It was a patchwork quilt in which each square consisted of separate arguments.
Moreover, the evidence only confirmed the original hypothesis, which involved making simplifying assumptions about the electron’s environment. More realistic situations are more messy: atoms in a solid are arranged in more convoluted patterns, and magnetic fields are not completely constant. “You verified it for this one model, but what does it have to do with reality?” he said Szymon Beckermathematician at the Swiss Federal Institute of Technology in Zurich.
These more realistic situations require adjustments to the alpha portion of the Schrödinger equation. And when you do that, the 10-martini proof stops working. “It was always disturbing to me,” Jitomirskaya said.
Breaking down the evidence into these broader contexts also suggested that the stunning fractal patterns that emerged – Cantor sets, Hofstadter’s butterfly – were nothing more than a mathematical curiosity, something that would disappear as the equation became more realistic.
Avila and Jitomirskaya moved on to other problems. Even Hofstadter had doubts. If his butterfly was ever seen in an experiment, he signed up Godel, Escher, Bach“I would be the most surprised person in the world.”
But in 2013, a group of physicists from Columbia University he captured his butterfly in the laboratory. They placed two lean layers of graphene in a magnetic field and then measured the energy levels of the graphene’s electrons. The quantum fractal has emerged in all its glory. “Suddenly everything changed from a figment of a mathematician’s imagination to something practical,” Jitomirskaja said. “It became very disturbing.”