Bootstrapping Heisenberg Magnets and their Cubic Instability

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 $\varphi$, and rank-2 symmetric tensor $t$. We determine their scaling dimensions to be $({\Delta }_s,{\Delta }_{\varphi },{\Delta }_t)=(0.518942(51),1.59489(59),1.20954(23))$, 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.

Comments: 35 pages, 9 figures, 8 tables. v2: clearing typos