On the Wilson-Fisher fixed point in the limit of integer spacetime dimensions

TL;DR

This paper investigates the relationship between the Wilson-Fisher fixed point and the Ising conformal field theory in integer dimensions, proposing that the Ising model is a subsector rather than identical to the fixed point, with implications for dimensional expansions.

Contribution

It challenges the assumption of a direct equivalence between the Wilson-Fisher fixed point and the Ising CFT at integer dimensions, proposing a subsector relationship and analyzing operator representations.

Findings

The Ising model emerges as a subsector of the Wilson-Fisher fixed point.

Operator multiplicities become negative at integer dimensions, indicating a nuanced relationship.

The results impact the understanding of $d=2+psilon$ expansions from two-dimensional data.

Abstract

The Wilson-Fisher fixed point defines a continuous family of interacting conformal field theories in non-integer dimensions. In integer dimensions, it is widely believed to lie in the same universality class as the critical Ising model. In this work, we revisit the identification between the Wilson-Fisher fixed point at integer dimensions and the Ising CFT. We argue that a literal equality between the two theories is incompatible with the emergence of Virasoro symmetry in two dimensions. Instead, we propose that the Ising model emerges only as a subsector of the Wilson-Fisher fixed point. We support this scenario through a detailed study of the two-dimensional $O(n)$ model and by examining operators transforming in irreducible representations of the orthogonal group whose multiplicities become negative for integer values of the spacetime dimension. Finally, we comment on the implications of these results for attempts to construct a $d=2+\epsilon$ expansion starting from exact two-dimensional data.