Scale invariance implies conformal invariance for the three-dimensional Ising model

TL;DR

This paper demonstrates that for the three-dimensional Ising model, scale invariance necessarily leads to conformal invariance, using Wilson renormalization group and Lebowitz inequalities.

Contribution

It proves that the absence of a certain vector operator ensures conformal invariance in the Ising universality class across all dimensions.

Findings

Scale invariance implies conformal invariance in the 3D Ising model.

The necessary condition is fulfilled in all dimensions for the Ising universality class.

The proof uses Wilson renormalization group and Lebowitz inequalities.

Abstract

Using Wilson renormalization group, we show that if no integrated vector operator of scaling dimension $-1$ exists, then scale invariance implies conformal invariance. By using the Lebowitz inequalities, we prove that this necessary condition is fulfilled in all dimensions for the Ising universality class. This shows, in particular, that scale invariance implies conformal invariance for the three-dimensional Ising model.