Exploring thermal order in conformal theories with multiple scalars coupled to an $O(N)$ vector field
We study thermal order in conformal field theories (CFTs) in $d=4-ε$ and $d=3$ dimensions where several scalars are coupled to an $O(N)$ vector field. In $(4-ε)$ dimensions, we consider models coupling a cubic model of $M$ scalars with $\mathbb{Z}_2^M\rtimes S_M$ symmetry (corresponding to sign flips and permutations of the scalars) to an $O(N)$ vector model. For any $M$, we find a window of $N$ in which two Wilson-Fisher-like fixed points exist. We show that the $\mathbb{Z}_2^M\rtimes S_M$ symmetry is spontaneously broken at arbitrary nonzero temperatures for $M=2$ and sufficiently large $N$ within this window, but it remains unbroken for $M>2$. Using the functional renormalisation group to continue these fixed points towards three dimensions, we find that they collide and move into the complex plane well before reaching $d=3$, suggesting that their continuation to $d=3$ does not yield unitary CFTs. We then work directly in three dimensions at large but finite $N$, with $M\ll N$, initially without assuming any permutation symmetry among the $M$ scalars. By studying the RG flow, we show that these models possess a conformal manifold up to the leading nontrivial order in the $1/N$ expansion of the beta functions (which is $\mathcal{O}(1/N)$). At each point on this manifold, the scalars split into two classes that are distinguished by how they couple to the $O(N)$ vector field. We then restrict to a subspace of the conformal manifold where there is an additional symmetry under permutations of the scalars within each class. We prove that in a domain of this subspace, all $M$ scalars acquire thermal expectation values such that the symmetry under sign flips of the scalars is spontaneously broken at all nonzero temperatures. This provides a novel class of large $N$ CFTs that exhibit a rich pattern of persistent symmetry breaking at nonzero temperatures.