A model of persistent breaking of discrete symmetry

We show there exist UV-complete field-theoretic models in general dimension, including $2+1$, with the spontaneous breaking of a global symmetry, which persists to the arbitrarily high temperatures. Our example is a conformal vector model with the $O(N)\times \mathbb{Z}_2$ symmetry at zero temperature. Using conformal perturbation theory we establish $\mathbb{Z}_2$ symmetry is broken at finite temperature for $N>10$. Similar to recent constructions, in the infinite $N$ limit our model has a non-trivial conformal manifold, a moduli space of vacua, which gets deformed at finite temperature. Furthermore, in this regime the model admits a persistent breaking of $O(N)$ in $2+1$ dimensions, therefore providing another example where the Coleman-Hohenberg-Mermin-Wagner theorem can be bypassed.