Bootstrapping Heisenberg Magnets and their Cubic Instability

TL;DR

This paper employs the numerical conformal bootstrap to precisely determine scaling dimensions and OPE bounds in the critical O(3) model, providing evidence of cubic anisotropy instability in related physical systems.

Contribution

It introduces a cutting-surface algorithm and a tip-finding method to map out CFT data and bound tensor operators in the critical O(3) model.

Findings

Determined scaling dimensions of key operators with high precision.

Bounded the leading rank-4 tensor operator, showing relevance.

Proved the instability of the critical O(3) model to cubic anisotropy.

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.