  In this work we explore the possibility of spontaneous breaking of global
symmetries at all nonzero temperatures for conformal field theories (CFTs) in
$D = 4$ space-time dimensions. We show that such a symmetry-breaking indeed
occurs in certain families of non-supersymmetric large $N$ gauge theories at a
planar limit. We also show that this phenomenon is accompanied by the system
remaining in a persistent Brout-Englert-Higgs (BEH) phase at any temperature.
These analyses are motivated by the work done in arXiv:2005.03676 where
symmetry-breaking was observed in all thermal states for certain CFTs in
fractional dimensions.
  In our case, the theories demonstrating the above features have gauge groups
which are specific products of $SO(N)$ in one family and $SU(N)$ in the other.
Working in a perturbative regime at the $N\rightarrow\infty$ limit, we show
that the beta functions in these theories yield circles of fixed points in the
space of couplings. We explicitly check this structure up to two loops and then
present a proof of its survival under all loop corrections. We show that under
certain conditions, an interval on this circle of fixed points demonstrates
both the spontaneous breaking of a global symmetry as well as a persistent BEH
phase at all nonzero temperatures. The broken global symmetry is $\mathbb{Z}_2$
in one family of theories and $U(1)$ in the other. The corresponding order
parameters are expectation values of the determinants of bifundamental scalar
fields in these theories. We characterize these symmetries as baryon-like
symmetries in the respective models.
