Bootstrapping Mixed Correlators in the 3D Ising Model

TL;DR

This paper extends the conformal bootstrap approach to mixed correlators in 3D CFTs with a $Z_2$ symmetry, providing numerical bounds on operator dimensions consistent with the 3D Ising model.

Contribution

It introduces a formalism for analyzing mixed correlators using semidefinite programming in 3D CFTs with $Z_2$ symmetry, yielding new bounds on operator dimensions.

Findings

Obtained a small allowed region for $(_, _)$ consistent with the 3D Ising model.

Demonstrated the effectiveness of mixed correlator bootstrap constraints.

Provided numerical bounds that narrow down the operator spectrum in 3D Ising CFT.

Abstract

We study the conformal bootstrap for systems of correlators involving non-identical operators. The constraints of crossing symmetry and unitarity for such mixed correlators can be phrased in the language of semidefinite programming. We apply this formalism to the simplest system of mixed correlators in 3D CFTs with a $\mathbb{Z}_2$ global symmetry. For the leading $\mathbb{Z}_2$-odd operator $\sigma$ and $\mathbb{Z}_2$-even operator $\epsilon$, we obtain numerical constraints on the allowed dimensions $(\Delta_\sigma, \Delta_\epsilon)$ assuming that $\sigma$ and $\epsilon$ are the only relevant scalars in the theory. These constraints yield a small closed region in $(\Delta_\sigma, \Delta_\epsilon)$ space compatible with the known values in the 3D Ising CFT.