'Ten Martini' Proof Uses Number Theory to Explain Quantum Fractals

Summary

But in some ways, the proof was a bit unsatisfying. Jitomirskaya and Avila had used a method that only applied to certain irrational values of alpha. By combining it with an intermediate proof that came before it, they could say the problem was solved. But this combined proof wasn’t elegant. It was a patchwork quilt, each square stitched out of distinct arguments. Moreover, the proofs only settled the conjecture as it was originally stated, which involved making simplifying assumptions about the electron’s environment. More realistic situations are messier: Atoms in a solid are arranged in more complicated patterns, and magnetic fields aren’t quite constant. “You’ve verified it for this one model, but what does that have to do with reality?” said Simon Becker, a mathematician at the Swiss Federal Institute of Technology Zurich. These more realistic situations require you to tweak the part of the Schrödinger equation where alpha appears. And when you do, the ten martini proof stops working. “This was always disturbing to me,” Jitomirskaya said. The breakdown of the proof in these broader contexts also implied that the beautiful fractal patterns that had emerged — the Cantor sets, the Hofstadter butterfly — were nothing more than a mathematical curiosity, something that would disappear once the equation was made more realistic. Avila and Jitomirskaya moved on to other problems. Even Hofstadter had doubts. If an experiment ever saw his butterfly, he’d written in Gödel, Escher, Bach, “I would be the most surprised person in the world.” But in 2013, a group of physicists at Columbia University captured his butterfly in a lab. They placed two thin layers of graphene in a magnetic field, then measured the energy levels of the graphene’s electrons. The quantum fractal emerged in all its glory. “Suddenly it went from a figment of the mathematician’s imagination to something practical,” Jitomirskaya said. “It became very unsettling.” She wanted to explain it with mathematics....