  Using the large-charge expansion, we prove a necessary condition for a CFT to
exhibit conformal symmetry breaking, under the assumption that a continuous
global symmetry is ${\it also}$ broken on the moduli space: there must be a
tower of charged local operators whose scaling dimensions are asymptotically
linear in the charge. In supersymmetric theories with a continuous R-symmetry
and a holomorphic moduli space, the existence of such a tower of operators
follows trivially from a BPS condition: their scaling dimensions are then
exactly linear in the R-charge. We illustrate the more general statement in
several examples of three-dimensional ${\cal N}=1$ CFTs, where the leading
linear behavior receives nontrivial corrections. By considering a suitable
scaling limit, we also relate the spectrum of states with large charge on the
cylinder (isomorphic to local operators) to the spectrum of massive particles
on the moduli space.
