In a spin chain, that motion is waves like those seen at stadiums. Flip one atom, and it will flip its neighbors, who will flip their neighbors, and so on. These waves are still extremely complicated; when two waves ripple through the same stretch of particles, any particle can flip any other particle, leading to pandemonium. Bloch’s theory forbade this messiness. He assumed that each atom could flip only its immediate neighbor. Then he guessed that, as a consequence, the resulting waves would always collide gently, passing through each other with minimal disruption. The assumption kept things tidy enough to handle.
Hans Bethe, a famously careful physicist, built on Bloch’s work to develop a way of perfectly capturing the behavior of certain complex quantum systems.
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His intuition nearly solved the problem, but he had overlooked one key mathematical detail. “Had he been less lazy with arctangents or logarithms, we would call this the Bloch ansatz,” said Jean-Sébastien Caux, a professor at the University of Amsterdam.
Bethe realized there was a second possibility for how two waves might coexist: They could attract each other so that they travel together. With that, Bethe captured everything the spin chain could do. Accounting for these two collective motions — gentle collisions and traveling in pairs — he could calculate the exact energy for every possible arrangement of the chain.
Bethe had stumbled upon a perfect quantum theory, one that worked for any number of particles. He never did use it to explain real-world magnets, however. It worked for chains but not for blocks of atoms as he had envisioned. Instead, it would prove its worth in other ways.
The Roots of Quantum Perfection
As Hitler rose to power in the following years, Bethe fled Germany, eventually reaching the United States and serving as a leader in the Manhattan Project. After the war he continued to study physics. But he never returned to his ansatz.
Others would discover just how much Bethe’s ansatz could do. It worked for spin chains with blemishes, and even for strings of particles influencing each other in non-magnetic ways. Puzzlingly, though, it continued to fail with the real-world blocks of atoms that had originally motivated Bethe. It wasn’t until the 1960s, when theorists applied it to thin sheets of ice — another system of countless quantum particles — that they figured out why.
Experimenters, cooling ice to unprecedentedly low temperatures, uncovered a mystery. If ice lost all its heat, they expected its molecules to settle down into a perfect, one-of-a-kind crystal. Instead, they found a strange disorder, as if molecules could end up in different arrangements that varied subtly from experiment to experiment.
Theorists realized that the frozen sheets also contained waves traveling down a line. Each sheet did indeed form a perfect crystal of repeating H2O molecules. But each molecule could take on one of six different configurations, like a pixel that can be either red, green, blue, yellow, orange or purple. Each time experimenters cooled down the ice, they got a different multicolored image. But there was a method in the madness. Theorists broke apart the image, starting at the top, taking it line by line. They treated each string of pixels like a frame in a movie. And when they played the movie, they saw waves. A green pixel might ripple down the line to the right, to give an overly simplistic example. And when these waves collided, they did so gently, keeping their shape, just as in Bethe’s spin chain.
So with the Bethe ansatz, physicists could precisely calculate the chances of measuring those patterns in an experiment. It was another perfect quantum theory.