(draft)

Most CFTs are strongly coupled, which means that there is no useful Lagrangian description. However, if the theory has some global symmetry group, and if we consider it in the sector with large associated charges, the effective theory describing those operators is effectively at weak coupling. More precisely, the effective coupling is the original coupling, but suppressed by powers of the global charge. See for instance here.

In the seminal paper by Hellerman, Orlando, Reffert and Watanabe, summarized here, the operator-state correspondence in $d$-dimensional CFT is used to map a scalar operator with $U(1)$ charge $Q$ to a state with homogeneous charge density in the theory conpactified on the cylinder $\mathbb{R} \times S^{d-1}$. If the radius of the sphere is $R$, the state has charge density $\rho \sim Q/R^{d-1}$. If the charge is large, $\rho^{1/(d-1)} >> 1/R$, and the CFT state and its excitations correspond to some "condensed matter phase". The configuration of Goldstone bosons in this case suggests that this phase should be called a "superfluid phase".