  It is widely expected that at sufficiently high temperatures order is always
lost, e.g. magnets loose their ferromagnetic properties. We pose the question
of whether this is always the case in the context of quantum field theory in
$d$ space dimensions. More concretely, one can ask whether there exist critical
points (CFTs) which break some global symmetry at arbitrary finite temperature.
The most familiar CFTs do not exhibit symmetry breaking at finite temperature,
and moreover, in the context of the AdS/CFT correspondence, critical points at
finite temperature are described by an uncharged black brane which obeys a
no-hair theorem. Yet, we show that there exist CFTs which have some of their
internal symmetries broken at arbitrary finite temperature. Our main example is
a vector model which we study both in the epsilon expansion and arbitrary rank
as well as the large rank limit (and arbitrary dimension). The large rank limit
of the vector model displays a conformal manifold, a moduli space of vacua, and
a deformed moduli space of vacua at finite temperature. The appropriate
Nambu-Goldstone bosons including the dilaton-like particle are identified.
Using these tools we establish symmetry breaking at finite temperature for
finite small $\epsilon$. We also prove that a large class of other fixed
points, which describe some of the most common quantum magnets, indeed behave
as expected and do not break any global symmetry at finite temperature. We
discuss some of the consequences of finite temperature symmetry breaking for
the spectrum of local operators. Finally, we propose a class of fixed points
which appear to be possible candidates for finite temperature symmetry breaking
in $d=2$.
