Bootstrapping the O(N) Vector Models

TL;DR

This paper uses the conformal bootstrap method to derive bounds on operator dimensions and central charges in 3D O(N) vector models, providing numerical evidence that these models saturate the bootstrap constraints across all N.

Contribution

The study provides the first rigorous bounds on operator dimensions and central charges in 3D O(N) models, confirming their saturation of bootstrap constraints and aligning with large-N expansion results.

Findings

Bounds on scaling dimensions match known critical exponents.

Central charge bounds agree with 1/N expansion at large N.

Evidence that O(N) models saturate bootstrap constraints at all N.

Abstract

We study the conformal bootstrap for 3D CFTs with O(N) global symmetry. We obtain rigorous upper bounds on the scaling dimensions of the first O(N) singlet and symmetric tensor operators appearing in the $\phi_i \times \phi_j$ OPE, where $\phi_i$ is a fundamental of O(N). Comparing these bounds to previous determinations of critical exponents in the O(N) vector models, we find strong numerical evidence that the O(N) vector models saturate the bootstrap constraints at all values of N. We also compute general lower bounds on the central charge, giving numerical predictions for the values realized in the O(N) vector models. We compare our predictions to previous computations in the 1/N expansion, finding precise agreement at large values of N.