However in some methods, the proof was a bit unsatisfying. Jitomirskaya and Avila had used a way that solely utilized to sure irrational values of alpha. By combining it with an intermediate proof that got here earlier than it, they may say the issue was solved. However this mixed proof wasn’t elegant. It was a patchwork quilt, every sq. stitched out of distinct arguments.
Furthermore, the proofs solely settled the conjecture because it was initially said, which concerned making simplifying assumptions concerning the electron’s setting. Extra reasonable conditions are messier: Atoms in a strong are organized in additional sophisticated patterns, and magnetic fields aren’t fairly fixed. “You’ve verified it for this one mannequin, however what does that should do with actuality?” mentioned Simon Becker, a mathematician on the Swiss Federal Institute of Know-how Zurich.
These extra reasonable conditions require you to tweak the a part of the Schrödinger equation the place alpha seems. And once you do, the 10-martini proof stops working. “This was all the time disturbing to me,” Jitomirskaya mentioned.
The breakdown of the proof in these broader contexts additionally implied that the gorgeous fractal patterns that had emerged—the Cantor units, the Hofstadter butterfly—had been nothing greater than a mathematical curiosity, one thing that may disappear as soon as the equation was made extra reasonable.
Avila and Jitomirskaya moved on to different issues. Even Hofstadter had doubts. If an experiment ever noticed his butterfly, he’d written in Gödel, Escher, Bach, “I might be probably the most stunned particular person on the earth.”
However in 2013, a gaggle of physicists at Columbia College captured his butterfly in a lab. They positioned two skinny layers of graphene in a magnetic discipline, then measured the power ranges of the graphene’s electrons. The quantum fractal emerged in all its glory. “All of the sudden it went from a figment of the mathematician’s creativeness to one thing sensible,” Jitomirskaya mentioned. “It grew to become very unsettling.”