Persistent Spontaneous Symmetry Breaking at High Temperatures

I will describe the results of a series of works in which we have found systems for which Spontaneous Symmetry Breaking(SSB) persists to very high temperatures. 
The systems involve Conformal Field Theories of scalars in fractional dimensions 3<d<4-\epsilon and for Gauge Theories in d=4.
The systems are studied in the limit of various large numbers of species. In the process we have found non supersymmetric
conformal manifolds as well as cases of non supersymmetric effective potentials with flat directions. The SSB results in the formation of a dilaton. We also discuss  some phase diagrams in the presence of relevant and marginally relevant operators made out of scalar fields in d=4.