Navigating through the O(N) archipelago

TL;DR

This paper extends the navigator method to explore the space of d-dimensional O(N) models, providing estimates of scaling dimensions across a broad parameter range and demonstrating the method's effectiveness in navigating complex conformal data landscapes.

Contribution

It applies the navigator method to the family of O(N) models in various dimensions, introducing a path-following algorithm for efficient exploration in the (d,N) plane.

Findings

Estimated scaling dimensions for (d,N) in [3,4] x [1,3]

Cannot detect non-unitary features due to fractional d or N

No solutions found for N<1 in the unitary crossing equations

Abstract

A novel method for finding allowed regions in the space of CFT-data, coined navigator method, was recently proposed in arXiv:2104.09518. Its efficacy was demonstrated in the simplest example possible, i.e. that of the mixed-correlator study of the 3D Ising Model. In this paper, we would like to show that the navigator method may also be applied to the study of the family of $d$-dimensional $O(N)$ models. We will aim to follow these models in the $(d,N)$ plane. We will see that the "sailing" from island to island can be understood in the context of the navigator as a parametric optimization problem, and we will exploit this fact to implement a simple and effective path-following algorithm. By sailing with the navigator through the $(d,N)$ plane, we will provide estimates of the scaling dimensions $(\Delta_{\phi},\Delta_{s},\Delta_{t})$ in the entire range $(d,N) \in [3,4] \times [1,3]$. We will show that to our level of precision, we cannot see the non-unitary nature of the $O(N)$ models due to the fractional values of $d$ or $N$ in this range. We will also study the limit $N \xrightarrow[]{} 1$, and see that we cannot find any solution to the unitary mixed-correlator crossing equations below $N=1$.