Thermal Order in the Biconical Model

Thermal fluctuations are generally expected to destroy order and restore symmetries at sufficiently high temperatures. Recently, however, a family of scalar theories in $2+1$ dimensions was shown to exhibit $\mathbb Z_2$ symmetry breaking that persists to arbitrarily high temperatures using a variety of approaches, including the $\epsilon$-expansion, the FRG, and large-$N$ techniques. Although the similarities among these theories suggest that they belong to the same universality class, this connection has not been established explicitly. In this work, we fill this gap. Using large-$N$ methods, we analytically study the biconical vector model in a range of spacetime dimensions, including $2+1$, at leading and next-to-leading order in the $1/N$ expansion. We determine the RG flow, fixed-point structure, and the CFT data of the infrared theory, reproducing and extending previous results and thereby unifying the apparently distinct constructions within a common analytic framework. We derive the effective potential, establish its stable minima at zero and finite temperature, and demonstrate spontaneous $\mathbb{Z}_2$ symmetry breaking at arbitrarily high temperatures for large finite $N$.