Exact critical exponents for vector operators in the 3d Ising model and conformal invariance

TL;DR

This paper derives exact critical exponents for vector operators in the 3d Ising model, confirming conformal invariance implications and identifying operators with fixed scaling dimensions across dimensions.

Contribution

It provides exact expressions for vector operator exponents in the 3d Ising model, demonstrating fixed scaling dimensions and their relevance to conformal invariance.

Findings

One operator has scaling dimension exactly 3 in any space dimension.

The operator considered in simulations also has scaling dimension exactly 3.

The results support the realization of conformal invariance at criticality.

Abstract

It is widely expected that the realization of scale invariance in the critical regime implies conformal invariance for a large class of systems. This is known to be true if there exist no integrated operator which transforms like a vector under rotations and which has scaling dimension $-1$. In this article we give exact expressions for the critical exponents of some of these vector operators. In particular, we show that one operator has scaling dimension exactly 3 in any space dimension. This operator turns out be the leading operator at least in $d=2$ and $d=4$. Moreover, we prove that the operator previously considered in Monte-Carlo simulations has also scaling dimension exactly $3$ in any dimension.