Abstract
We study the critical $O(3)$ model using the numerical conformal bootstrap. In particular, we use a recently developed cutting-surface algorithm to efficiently map out the allowed space of CFT data from correlators involving the leading $O(3)$ singlet $s$, vector $\phi$, and rank-2 symmetric tensor $t$. We determine their scaling dimensions to be $(\Delta_{s}, \Delta_{\phi}, \Delta_{t}) = (0.518942(51), 1.59489(59), 1.20954(23))$, and also bound various OPE coefficients. We additionally introduce a new "tip-finding" algorithm to compute an upper bound on the leading rank-4 symmetric tensor $t_4$, which we find to be relevant with $\Delta_{t_4} < 2.99056$. The conformal bootstrap thus provides a numerical proof that systems described by the critical $O(3)$ model, such as classical Heisenberg ferromagnets at the Curie transition, are unstable to cubic anisotropy.
Highlights
Numerical bootstrap methods [1,2] have led to powerful new results in the study of conformal field theories (CFTs)
In Refs. [5,6] we developed an approach to large-scale bootstrap problems which allowed for precise determinations of the CFT data of the 3D critical Oð2Þ model
Exploration of large-scale bootstrap problems by applying the technology introduced in Ref. [5] to the study of the 3D critical Oð3Þ model
Summary
Numerical bootstrap methods [1,2] (see Refs. [3,4] for recent reviews) have led to powerful new results in the study of conformal field theories (CFTs). [5,6] we developed an approach to large-scale bootstrap problems which allowed for precise determinations of the CFT data of the 3D critical Oð2Þ model. Exploration of large-scale bootstrap problems by applying the technology introduced in Ref. We introduce a new algorithm and software implementation called tiptop, which allows us to efficiently test allowed gaps for other operators across this region.
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