Mathematics is far more fraught with debate and disagreement than you might imagine. Arguments about things some of the smartest physicists have trouble understanding rage for years. Recently, a pair of mathematicians ignited some old flames — or rather, shattered some glass — with a new set of results that, if correct, have far-reaching implications in physics and even cybersecurity.

Image: Flickr/H.Adam

Duke physics and maths postdoc Sho Yaida and his adviser Patrick Charbonneau published a new paper in *Physical Review Letters* finding that the structure of glass on an atomic level can be even stranger than we thought. But the calculation could have importance across disordered systems, structures including, as one paper says, "liquids and glasses, grains and foams, galaxies, etc". When Yaida told Charbonneau he wanted to work on this problem — figuring out whether something called "replica symmetry breaking" occurred in glasses with less than six dimensions — Charbonneau knew they were entering rocky waters.

"We were walking to a meeting together and while we were talking, Sho said, 'I think I have an idea for how to solve this problem you didn't tell me to work on.' During the whole meeting I wondered if I misunderstood," Charbonneau told Gizmodo. "I knew this could cause trouble. I didn't know to what extent. I knew the reputation of the problem but didn't know the personalities. I didn't know who was going to be happy, who was going to be convinced, and who would push back."

The problem concerns something called replica symmetry breaking in glass, which according to Yaida is only "well established in infinite dimensions".

"But we only have three-dimensional space," he said. "For 30 years people wondered about how this happens in three-dimensional materials."

I assume you are familiar with the debate surrounding replica symmetry breaking in disordered systems in less than six dimensions. No? Fine, I will explain it.

The debate centres on the complex mathematics describing arrangements of things (atoms, for example) that are random but retain a memory of their initial configuration. Glass is what Charbonneau and Yaida are specifically interested in — it's kind of like super-thick liquid whose random atomic structure keeps its shape. The pair wanted to know how its atomic structure changes under the influence of things like breaking, pressure or temperature.

The two simplify these kinds of problems and abstract the hell out of them, looking at them in any number of dimensions. Mathematicians aren't limited by three spatial dimensions (technically, neither are physicists) and abstracting these problems to more dimensions can make them applicable to non-physical things, like computing algorithms.

Researchers studying these many-dimension disordered systems have long wondered whether this replica symmetry breaking occurs below a magic number of dimensions, six. Essentially, with a certain combination of variables, let's say temperature and pressure, the individual parts of the system, let's say the atoms in the glass, would no longer have a single most likely configuration, as it usually the case when a liquid freezes to a solid. Instead, the material could have several almost-most likely configurations. If the replica symmetry breaks in glass, then the atoms take on different configurations in different parts of the glass. It's kind of like moving from filling the back of a car with a lot of same-sized bricks to filling it with differently awkwardly-sized pieces of luggage — there's only one arrangement in the first example, but lots of possible arrangements in the second.

If this is hard to understand, that's because it's incredibly complex. Several of the smartest physicists and mathematicians I emailed working in adjacent fields said the maths went over their heads (same, by the way). One just said he was unfamiliar with the problem but that Yaida is a good physicist.

Anyway, Charbonneau and Yaida's research found evidence that this replica symmetry breaking occurs in real-world glasses we actually care about, those in three dimensions. And that's important because it can affect lots of glasses' properties. But the finding is controversial. Others had begun this calculation several decades ago and concluded that replica symmetry breaking probably wouldn't happen. Yaida essentially plugged that 30-year-old algorithm back into itself and turn the mathematical crank a second time — and his results implied a different conclusion. Emeritus Professor Michael Moore at the University of Manchester, who's been working on the problem in another system, told the two that they were completely incorrect.

The controversy comes from the fact that Charbonneau and Yaida's solution for regular glasses should also work for other disordered systems, like spin glasses. Spin glasses aren't glasses at all, but metals whose atoms' spin (an innate physical property which either equals "up" or "down") is all disordered. Yaida and Charbonneau's calculation essentially says that replica symmetry breaking occurs in spin glasses too.

If that's true, it would have lots of implications for spin glasses. Spin glasses are real, but don't have many useful physical applications. Instead, the mathematics that describes them is incredibly useful. There are a whole lot of concepts in computing that might use the same maths, including artificial neural networks. It could even help the CIA or FBI crack encryption keys. This maths is really important.

Moore is saying that Yaida's maths doesn't conclusively show that the replica symmetry breaking happens in less than six dimensions. "I don't think the paper is wrong. It's just very speculative," he told Gizmodo. He and his collaborators have a new paper about spin glasses currently in review that he thinks refutes Yaida and Charbonneau's. "I wouldn't claim ours is rigorous either. But it indicates that it's unlikely that their scenario is correct."

Moore would like to see the mathematical crank turned a third time, as well as a more rigorous proof that doesn't involve continuing to turn a crank to get better and better approximations.

Other researchers had different feelings on Yaida's maths, but the general sentiment seemed to be that it was a good direction to take the problem. "I think that it's more than speculative," M. Lisa Manning, associate professor at Syracuse University, told Gizmodo. "But there's still some wiggle room for things to change. It's the right next step in my opinion to pinning the solution down. A lot of us didn't expect that a two-loop calculation," turning the crank a second time, "would give this result." Researcher Giorgio Parisi has been working on this problem for a long time and agreed with that assessment. "It's a first step in that direction, but one needs to look better in that direction."

Helmut Katzgraber, professor at Texas A&M University, is working with Moore on his new paper and felt the same way as his coauthor — that ultimately, while impressive, these calculations might change drastically if they're re-run a third time. He thinks the problem has been over-chewed. But he stands by the idea that there's no replica symmetry breaking in less than six dimensions.

Charbonneau thought that maybe Moore and Katzgraber were biased, because spin glasses are different from regular glasses. Maybe the symmetry breaking occurs in some way that hasn't been observed yet. While he thinks Yaida's maths pretty conclusively demonstrates the presence of the broken symmetry, he knows that the story isn't quite over yet.

"When we submitted this work we thought we made a really major step," he said. "But the back-and-forth will probably last a few months to a few years before the steady state of understanding is reached."