Bootstrapping Heisenberg magnets and their cubic instability

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 conformal field theory data from correlators involving the leading O(3) singlet s, vector φ, and rank-2 symmetric tensor t. We determine their scaling dimensions to be (Δφ,Δs,Δt)=(0.518942(51),1.59489(59),1.20954(23)), and also bound various operator product expansion coefficients. We additionally introduce a new "tip-finding"algorithm to compute an upper bound on the leading rank-4 symmetric tensor t4, which we find to be relevant with Δt4<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.