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Transition of Large $R$-Charge Operators on a Conformal Manifold

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