Mathematician proves the symmetry of phase transition

The existence of conformal invariance has direct physical meaning: it shows that even if you adjust the microscopic details of matter, the overall behavior of the system will not change. It also implies a certain mathematical elegance, in a brief episode, as if the whole system is breaking its overall form and becoming something else.

First proof

In 2001, Smirnov proposed the first strict mathematical prove The conformal invariance in the physical model. It is suitable for the penetration model, which is the process of liquid passing through a maze in a porous medium (such as stone).

Smirnov studied the penetration on a triangular lattice, in which water can only flow through the “open” vertices. Initially, each vertex has the same probability of being open to water flow. When the probability is low, the probability of water passing through the stone all the way is low.

But as you slowly increase the probability, there will be a point where enough vertices are open to create the first path across the stone. Smirnov proved that at the critical threshold, the triangular lattice is conformal invariant, which means that no matter how you transform it with conformal symmetry, penetration occurs.

Five years later, at the 2006 International Congress of Mathematicians, Smirnov Announce He once again proved the shape-preserving invariance, this time in the Ising model. Combined with his 2001 proof, this pioneering work won him the Fields Medal, the highest honor in mathematics.

In the years since then, other evidence has emerged case by case, establishing conformal invariance for a particular model. No one can prove the universality that Polyakov envisioned.

“The previous valid proof was tailored to a specific model,” Federico Camilla, A mathematical physicist at New York University Abu Dhabi. “You have a very specific tool to prove a very specific model.”

Smirnov himself admits that both of his proofs rely on some kind of “magic” that exists in the two models he uses but is usually not available.

“Because it uses magic, it only works when there is magic, and we can’t find magic in other situations,” he said.

This new work is the first work to break this pattern-it proves that rotation invariance, the core feature of conformal invariance, exists widely.

One at a time

Duminil-Copin first began to consider proving universal conformal invariance in the late 2000s, when he was a graduate student of Smirnov at the University of Geneva. He has a unique understanding of the brilliance of the mentor’s skills—and their limitations. Smirnov bypassed the need to prove all three symmetries separately, but found a direct way to establish conformal invariance—like a shortcut to the top.

“He is an amazing problem solver. He proved the conformal invariance of the two statistical physical models by finding the entrance on this huge mountain, just like the crux of his passing,” Duminil-Copin said.

In the years after graduating, Duminil-Copin worked hard to establish a set of proofs that might eventually allow him to surpass Smirnov’s work. When he and his co-authors began to study form-preserving invariance seriously, they were ready to adopt a different approach than Smirnov. They did not take risks with magic, but went back to the original hypothesis of conformal invariance put forward by Polyakov and later physicists.

Physicists need to prove in three steps that every symmetry has a symmetry: translation invariance, rotation invariance and scale invariance. Prove each of them separately, and you will get shape-preserving invariance.

With this in mind, the author first set out to prove scale invariance, thinking that rotation invariance would be the most difficult symmetry, and knew that translation invariance was simple enough to prove it by himself. In trying this, they realized that they could prove the existence of rotation invariance at the critical points of various seepage models on square and rectangular grids.