For more than 50 years, mathematicians have been searching for a rigorous way to prove that an unusually strong symmetry is universal across physical systems at the mysterious juncture where they’re changing from one state into another. The powerful symmetry, known as conformal invariance, is actually a package of three separate symmetries that are all wrapped up within it.
Now, in a proof posted in December, a team of five mathematicians has come closer than ever before to proving that conformal invariance is a necessary feature of these physical systems as they transition between phases. The work establishes that rotational invariance — one of the three symmetries contained within conformal invariance — is present at the boundary between states in a wide range of physical systems.
“It’s a major contribution. This was open for a long time,” said Gady Kozma of the Weizmann Institute of Science in Israel.
Rotational invariance is a symmetry exhibited by the circle: Rotate it any number of degrees and it looks the same. In the context of physical systems on the brink of phase changes, it means many properties of the system behave the same regardless of how a model of the system is rotated.
Earlier results had established that rotational invariance holds for two specific models, but their methods were not flexible enough to be used for other models. The new proof breaks from this history and marks the first time that rotational invariance has been proved to be a universal phenomenon across a broad class of models.
“This universality result is even more intriguing” because it means that the same patterns emerge regardless of the differences between models of physical systems, said Hugo Duminil-Copin of the Institute of Advanced Scientific Studies (IHES) and the University of Geneva.
Duminil-Copin is a co-author of the work along with Karol Kajetan Kozlowski of the École Normale Supérieure in Lyon, Dmitry Krachun of the University of Geneva, Ioan Manolescu of the University of Fribourg and Mendes Oulamara of IHES and Paris-Saclay University.
The new work also raises hopes that mathematicians might be closing in on an even more ambitious result: proving that these physical models are conformally invariant. Over the last several decades mathematicians have proved that conformal invariance holds for a few particular models, but they’ve been unable to prove that it holds for all of them, as they suspect it does. This new proof lays the foundation for sweeping results along those lines.
“It’s already a very big breakthrough,” said Stanislav Smirnov of the University of Geneva. “[Conformal invariance] now looks within reach.”
Transitions between one state and another are some of the most mesmerizing events in the natural world. Some are abrupt, like the transformation of water when it heats into vapor or cools into ice. Others, like the phase transitions studied in the new work, evolve gradually, with a murky boundary between two states. It’s here, at these critical points, that the system hangs in the balance and is neither quite what it was nor what it’s about to become.
Mathematicians try to bottle this magic in simplified models.
Take, for example, what happens as you heat iron. Above a certain temperature it loses its magnetic attraction. The change occurs as millions of sizzling atoms acting as miniature magnets flip and no longer align with the magnetic positions of their neighbors. Around 1,000 degrees Fahrenheit, heat wins out and a magnet reduces to a mere piece of metal.
Mathematicians study this process with the Ising model. It imagines a block of iron as a two-dimensional square lattice, much like the grid on a piece of graph paper. The model situates the iron atoms at the intersections of the lattice lines and represents them as arrows pointing up or down.
The Ising model came into widespread use in the 1950s as a tool to represent physical systems near critical points. These included metals losing magnetism and also the gas-liquid transition in air and the switch between order and disorder in alloys. These are all very different types of systems that behave in very different ways at the microscopic level.
Then, in 1970, the young physicist Alexander Polyakov predicted that despite their apparent differences, these systems all exhibit conformal invariance at their critical points. Decades of subsequent analysis convinced physicists that Polyakov was right. But mathematicians have been left with the difficult job of rigorously proving that it’s true.
The Symmetry of Symmetries
Conformal invariance consists of three types of symmetries rolled into one more extensive symmetry. You can shift objects that exhibit it (translational symmetry), rotate them by any number of degrees (rotational symmetry or invariance), or change their size (scale symmetry), all without changing any of their angles.
“[Conformal invariance] is what sometimes I call ‘the symmetry to rule them all’ because it’s an overall symmetry, which is stronger than the three others,” said Duminil-Copin.
Conformal invariance shows up in physical models in a more subtle way. In the Ising model, when magnetism is still intact and a phase transition hasn’t occurred yet, most arrows point up in one massive cluster. There are also some small clusters in which all arrows point down. But at the critical temperature, atoms can influence each other from greater distances than before. Suddenly, the alignment of atoms everywhere is unstable: Clusters of different sizes with arrows pointing either up or down appear all at once, and magnetism is about to be lost.
At this critical point, mathematicians look at the model from very far away and study correlations between arrows, which characterize the likelihood that any given pair points in the same direction. In this setting, conformal invariance means that you can translate, rotate and rescale the grid without distorting those correlations. That is, if two arrows have a 50% chance of pointing in the same direction, and then you apply those symmetries, the arrows that come to occupy the same positions in the lattice will also have a 50% chance of aligning.
The result is that if you compare your original lattice model with the new, transformed lattice, you won’t be able to tell which is which. Importantly, the same is not true of the Ising model before the phase transition. There, if you take the top corner of the lattice and blow it up to be the same size as the original (a scale transformation), you’ll also increase the typical size of the small islands of down arrows, making it obvious which lattice is the original.
The presence of conformal invariance has a direct physical meaning: It indicates that the global behavior of the system won’t change even if you tweak the microscopic details of the substance. It also hints at a certain mathematical elegance that sets in, for a brief interlude, just as the entire system is breaking its overarching form and becoming something else.
The First Proofs
In 2001 Smirnov produced the first rigorous mathematical proof of conformal invariance in a physical model. It applied to a model of percolation, which is the process of liquid passing through a maze in a porous medium, like a stone.
Smirnov looked at percolation on a triangular lattice, where water is allowed to flow only through vertices that are “open.” Initially, every vertex has the same probability of being open to the flow of water. When the probability is low, the chances of water having a path all the way through the stone is low.
But as you slowly increase the probability, there comes a point where enough vertices are open to create the first path spanning the stone. Smirnov proved that at the critical threshold, the triangular lattice is conformally invariant, meaning percolation occurs regardless of how you transform it with conformal symmetries.
Five years later, at the 2006 International Congress of Mathematicians, Smirnov announced that he had proved conformal invariance again, this time in the Ising model. Combined with his 2001 proof, this groundbreaking work earned him the Fields Medal, math’s highest honor.
In the years since, other proofs have trickled in on a case-by-case basis, establishing conformal invariance for specific models. None have come close to proving the universality that Polyakov envisioned.
“The previous proofs that worked were tailored to specific models,” said Federico Camia, a mathematical physicist at New York University Abu Dhabi. “You have a very specific tool to prove it for a very specific model.”
Smirnov himself acknowledged that both of his proofs relied on some sort of “magic” that was present in the two models he worked with but isn’t usually available.
“Since it used magic, it only works in situations where there is magic, and we weren’t able to find magic in other situations,” he said.
The new work is the first to disrupt this pattern — proving that rotational invariance, a core feature of conformal invariance, exists widely.
One at a Time
Duminil-Copin first began to think about proving universal conformal invariance in the late 2000s, when he was Smirnov’s graduate student at the University of Geneva. He had a unique understanding of the brilliance of his mentor’s techniques — and also of their limitations. Smirnov bypassed the need to prove all three symmetries separately and instead found a direct route to establishing conformal invariance — like a shortcut to a summit.
“He’s an amazing problem solver. He proved conformal invariance of two models of statistical physics by finding the entrance in this huge mountain, like this kind of crux that he went through,” said Duminil-Copin.
For years after graduate school, Duminil-Copin worked on building up a set of proofs that might eventually allow him to go beyond Smirnov’s work. By the time he and his co-authors set to work in earnest on conformal invariance, they were ready to take a different approach than Smirnov had. Rather than take their chances with magic, they returned to the original hypotheses about conformal invariance made by Polyakov and later physicists.
The physicists had required a proof in three steps, one for each symmetry present in conformal invariance: translational, rotational and scale invariance. Prove each of them separately, and you get conformal invariance as a consequence.
With this in mind, the authors set out to prove scale invariance first, believing that rotational invariance would be the most difficult symmetry and knowing that translational invariance was simple enough and wouldn’t require its own proof. In attempting this, they realized instead that they could prove the existence of rotational invariance at the critical point in a large variety of percolation models on square and rectangular grids.
They used a technique from probability theory, called coupling, that made it possible to directly compare the large-scale behavior of square lattices with rotated rectangular lattices. By combining this approach with ideas from another field of mathematics called integrability, which studies hidden structures in evolving systems, they were able to prove that the behavior at critical points was the same across the models — thus establishing rotational invariance. Then they proved that their results extended to other physical models where it’s possible to apply the same coupling.
The end result is a powerful proof that rotational invariance is a universal property of a large subset of known two-dimensional models. They believe the success of their work indicates that a similarly eclectic set of techniques, melded from various fields of math, will be necessary to make additional progress on conformal invariance.
“I think it’s going to be more and more true, in arguments of conformal invariance and the study of phase transitions, that you need a little bit of everything. You cannot just attack it with one angle of attack,” said Duminil-Copin.
For the first time since Smirnov’s 2001 result, mathematicians have new purchase on the long-standing challenge of proving the universality of conformal invariance. And unlike that earlier work, this new result opens new paths to follow. By following a bottom-up approach in which they aimed to prove one constituent symmetry at a time, the researchers hope they laid a foundation that will eventually support a universal set of results.
Now, with rotational invariance down, Duminil-Copin and his colleagues have their sights set on scale invariance, their original target. A proof of scale invariance, given the recent work on rotational symmetry and the fact that translational symmetry doesn’t need its own proof, would put mathematicians on the cusp of proving full conformal invariance. And the flexibility of their methods makes the researchers optimistic it can be done.
“I definitely think that step three is going to fall fairly soon,” said Duminil-Copin. “If it’s not us, it would be somebody smarter, but definitely, it’s going to happen very soon.”
The proof of rotational invariance took five years, though, so the next results may yet take some time. Still, Smirnov is hopeful that two-dimensional conformal invariance may finally be within reach.
“That might mean a week, or it might mean five years, but I’m much more optimistic than I was in November,” said Smirnov.