Rigorous bounds on irrelevant operators in the 3d Ising model CFT

Abstract

We use the recently developed navigator method to obtain rigorous upper and lower bounds on new OPE data in the 3d Ising CFT. For example, assuming that there are only two ${\mathbb{Z}}_2$-even scalar operators $\epsilon$ and ${\epsilon }^{\prime }$ with a dimension below 6 we find a narrow allowed interval for ${\Delta }_{{\epsilon }^{\prime }}$, ${\lambda }_{\sigma \sigma {\epsilon }^{\prime }}$ and ${\lambda }_{\epsilon \epsilon {\epsilon }^{\prime }}$. With similar assumptions in the ${\mathbb{Z}}_2$-even spin-2 and the ${\mathbb{Z}}_2$-odd scalar sectors we are also able to constrain: the central charge $c_T$; the OPE data ${\Delta }_{T^{\prime }}$, ${\lambda }_{\epsilon \epsilon T^{\prime }}$ and ${\lambda }_{\sigma \sigma T^{\prime }}$ of the second spin-2 operator; and the OPE data ${\Delta }_{{\sigma }^{\prime }}$ and ${\lambda }_{\sigma \epsilon {\sigma }^{\prime }}$ of the second ${\mathbb{Z}}_2$-odd scalar. We compare the rigorous bounds we find with estimates that have been previously obtained using the extremal functional method (EFM) and find a good match. This both validates the EFM and shows the navigator-search method to be a feasible and more rigorous alternative for estimating a large part of the low-dimensional operator spectrum. We also investigate the effect of imposing sparseness conditions on all sectors at once. We find that the island does not greatly reduce in size under these assumptions. We efficiently find islands and determine their size in high-dimensional parameter spaces (up to 13 parameters). This shows that using the navigator method the numerical conformal bootstrap is no longer constrained to the exploration of small parameter spaces.

34 pages, 7 figures

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