SciPost Phys. 13, 081 (2022) ·
published 4 October 2022
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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 $N=1$.
Marten Reehorst, Slava Rychkov, David Simmons-Duffin, Benoit Sirois, Ning Su, Balt van Rees
SciPost Phys. 11, 072 (2021) ·
published 28 September 2021
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Current numerical conformal bootstrap techniques carve out islands in theory space by repeatedly checking whether points are allowed or excluded. We propose a new method for searching theory space that replaces the binary information “allowed”/“excluded” with a continuous “navigator” function that is negative in the allowed region and positive in the excluded region. Such a navigator function allows one to efficiently explore high-dimensional parameter spaces and smoothly sail towards any islands they may contain. The specific functions we introduce have several attractive features: they are well-defined in large regions of parameter space, can be computed with standard methods, and evaluation of their gradient is immediate due to an SDP gradient formula that we provide. The latter property allows for the use of efficient quasi-Newton optimization methods, which we illustrate by navigating towards the 3d Ising island.
