Universality for critical lines for Ising, Vertex and Dimer models

TL;DR

This paper reviews a rigorous proof that universality and scaling relations hold along critical lines in various planar lattice models, regardless of their solvability, using Renormalization Group methods.

Contribution

It provides a rigorous validation of universality and exact scaling relations for critical exponents in lattice models via Renormalization Group techniques.

Findings

Universality holds for critical exponents in these models.

Scaling relations are proven to be valid.

The proof is model-independent and relies on RG methods.

Abstract

In planar lattice statistical mechanics models like coupled Ising with quartic interactions, vertex and dimer models, the exponents depend on all the Hamiltonian details. This corresponds, in the Renormalization Group language, to a line of fixed points. A form of universality is expected to hold, implying that all the exponents can be expressed by exact "Kadanoff" relations in terms of a single one of them. This conjecture has been recently established and we review here the key step of the proof, obtained by rigorous Renormalization Group methods and valid irrespectively on the solvability of the model. The exponents are expressed by convergent series in the coupling and, thanks to a set of cancellations due to emerging chiral symmetries, the extended scaling relations are proven to be true.