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Mathematicians Prove Symmetry of Phase Transitions

The presence of conformal invariance has direct physical significance: it indicates that the overall behavior of the system will not change even if you change the microscopic details of the substance. It also alludes to a certain mathematical elegance that sets in, the time of a brief interlude, at the very moment when the whole system breaks its global form and becomes something else.

The first proofs

In 2001, Smirnov produced the first rigorous mathematical proof of conformal invariance in a physical model. He applied himself to a model of percolation, which is the process of passing a liquid through a labyrinth in a porous medium, like a stone.

Smirnov examined percolation on a triangular network, where water can only flow through “open” peaks. Initially, each vertex has the same probability of being open to the flow of water. When the probability is low, the chances of the water having a path through the stone are low.

But as you slowly increase the probability, there comes a point when enough peaks are open to create the first path covering the stone. Smirnov proved that at the critical threshold, the triangular lattice is conformally invariant, meaning that 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 again proved conformal invariance, this time in the Ising model. Combined with his 2001 ordeal, this groundbreaking work earned him the Fields Medal, the highest honor in mathematics.

In the years that followed, further evidence emerged on a case-by-case basis, establishing consistent invariance for specific models. None has succeeded in proving the universality envisaged by Polyakov.

“The previous evidence that worked was tailored to specific models,” said Federico Camia, a mathematical physicist at New York University in Abu Dhabi. “You have a very specific tool to prove it for a very specific model. “

Smirnov himself admitted that his two proofs were based on a kind of “magic” that was present in the two models he worked with, but which is generally not available.

“Since he was using magic, it only works in situations where there is magic, and we couldn’t find magic in other situations,” he said.

The new work is the first to disrupt this model, proving that rotational invariance, an essential feature of conformal invariance, widely exists.

One at a time

Duminil-Copin 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, as well as their limitations. Smirnov circumvented the need to prove the three symmetries separately and instead found a direct way to establish conformal invariance, such as a shortcut to a vertex.

“He’s a great problem solver. He proved the conformal invariance of two models of statistical physics by finding the entrance to this huge mountain, like this kind of node that he passed through, ”said Duminil-Copin.

For years after graduating from graduate school, Duminil-Copin worked to build up a body of evidence that could eventually allow him to go beyond Smirnov’s work. By the time he and his co-authors got down to serious work on conformal invariance, they were ready to take a different approach than Smirnov’s. Rather than try their luck with magic, they reverted to the original assumptions about conformal invariance made by Polyakov and later physicists.

Hugo Duminil-Copin of the Institut des Hautes Etudes Scientifique and the University of Geneva and his collaborators take a symmetry-at-a-time approach to prove the universality of conformal invariance.Photography: IHES / MC Vergne

Physicists had demanded 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 accordingly.

With this in mind, the authors began by proving scale invariance, believing that rotational invariance would be the most difficult symmetry, and knowing that translational invariance was quite simple and would not require its own proof. While trying this, they realized that they could prove the existence of rotational invariance at the critical point in a wide variety of percolation models on square and rectangular grids.



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