However in some methods, the proof was a bit unsatisfying. Jitomirskaya and Avila had used a technique that solely utilized to sure irrational values of alpha. By combining it with an intermediate proof that got here earlier than it, they might 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 acknowledged, which concerned making simplifying assumptions in regards to the electron’s surroundings. Extra practical conditions are messier: Atoms in a stable are organized in additional difficult patterns, and magnetic fields aren’t fairly fixed. “You’ve verified it for this one mannequin, however what does that should do with actuality?” stated Simon Becker, a mathematician on the Swiss Federal Institute of Expertise Zurich.
These extra practical conditions require you to tweak the a part of the Schrödinger equation the place alpha seems. And if you do, the 10-martini proof stops working. “This was at all times disturbing to me,” Jitomirskaya stated.
The breakdown of the proof in these broader contexts additionally implied that the attractive fractal patterns that had emerged—the Cantor units, the Hofstadter butterfly—had been nothing greater than a mathematical curiosity, one thing that will disappear as soon as the equation was made extra practical.
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 essentially the most shocked 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 subject, then measured the vitality ranges of the graphene’s electrons. The quantum fractal emerged in all its glory. “Instantly it went from a figment of the mathematician’s creativeness to one thing sensible,” Jitomirskaya stated. “It grew to become very unsettling.”
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