  We study the transition between phases at large $R$-charge on a conformal
manifold. These phases are characterized by the behaviour of the lowest
operator dimension $\Delta(Q_R)$ for fixed and large $R$-charge $Q_R$. We
focus, as an example, on the $D=3$, $\mathcal{N}=2$ Wess-Zumino model with
cubic superpotential $W=XYZ+\frac{\tau}{6}(X^3+Y^3+Z^3)$, and compute
$\Delta(Q_R,\tau)$ using the $\epsilon$-expansion in three interesting limits.
In two of these limits the (leading order) result turns out to be
\begin{equation*}
  \Delta(Q_R,\tau)=
  \begin{cases}
  \left(\text{BPS bound}\right)\left[1+O(\epsilon |\tau|^2Q_R)\right], & Q_R\ll
\left\{ \frac{1}{\epsilon},\, \frac{1}{\epsilon|\tau|^2}\right\}\\
\frac{9}{8}\left(\frac{\epsilon|\tau|^2}{2+|\tau|^2}\right)^{\frac{1}{D-1}}Q_R^{\frac{D}{D-1}}
  \left[1+O\left(\left(\epsilon
|\tau|^2Q_R\right)^{-\frac{2}{D-1}}\right)\right],
  & Q_R\gg \left\{ \frac{1}{\epsilon},\, \frac{1}{\epsilon|\tau|^2}\right\}
  \end{cases}
  \end{equation*} which leads us to the double-scaling parameter, $\epsilon
|\tau|^2Q_R$, which interpolates between the "near-BPS phase" ($\Delta(Q)\sim
Q$) and the "superfluid phase" ($\Delta(Q)\sim Q^{D/(D-1)}$) at large
$R$-charge. This smooth transition, happening near $\tau=0$, is a
large-$R$-charge manifestation of the existence of a moduli space and an
infinite chiral ring at $\tau=0$. We also argue that this behavior can be
extended to three dimensions with minimal modifications, and so we conclude
that $\Delta(Q_R,\tau)$ experiences a smooth transition around $Q_R\sim
1/|\tau|^2$. Additionally, we find a first-order phase transition for
$\Delta(Q_R,\tau)$ as a function of $\tau$, as a consequence of the duality of
the model. We also comment on the applicability of our result down to small
$R$-charge.
