Mathematicians Prove Symmetry of Phase Transitions | Quanta Magazine

Abstraction: Proof of universal rotational invariance at physical phase transition critical points

Key points:

• A five-mathematician team (led by Hugo Duminil-Copin, IHES/University of Geneva) proved that rotational invariance — one of three symmetries comprising conformal invariance — is universal across a broad class of two-dimensional physical models at their critical points
• Conformal invariance (predicted by Alexander Polyakov in 1970) packages translational, rotational, and scale symmetry; it describes systems at phase-transition critical points regardless of microscopic differences
• Prior proofs by Stanislav Smirnov (2001 percolation, 2006 Ising model) used model-specific "magic" and couldn't generalize; he earned the Fields Medal for those results
• The new proof uses coupling (probability theory) combined with integrability techniques to compare square and rotated rectangular lattices at critical points
• This is the first universality result: rotational invariance now proven for a wide class of models, not just specific ones
• The team is optimistic that scale invariance (the remaining step) will fall soon, completing a proof of full conformal invariance

Connections: Ihes · University Of Geneva · Conformal Invariance · Phase Transitions · Mathematical Physics

Source: https://www.quantamagazine.org/mathematicians-prove-symmetry-of-phase-transitions-20210708/