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 ∆(QR) for fixed and large R-charge QR. We focus, as an example, on the D = 3, $$ \mathcal{N} $$ N = 2 Wess-Zumino model with cubic superpotential $$ W= XYZ+\frac{\tau }{6}\left({X}^3+{Y}^3+{Z}^3\right) $$ W = XYZ + τ 6 X 3 + Y 3 + Z 3 , and compute ∆(QR, τ) using the ϵ-expansion in three interesting limits. In two of these limits the (leading order) result turns out to be$$ \Delta \left({Q}_{R,\tau}\right)=\left\{\begin{array}{ll}\left(\mathrm{BPS}\;\mathrm{bound}\right)\left[1+O\left(\epsilon {\left|\tau \right|}^2{Q}_R\right)\right],& {Q}_R\ll \left\{\frac{1}{\epsilon },\kern0.5em \frac{1}{\epsilon {\left|\tau \right|}^2}\right\}\\ {}\frac{9}{8}{\left(\frac{\epsilon {\left|\tau \right|}^2}{2+{\left|\tau \right|}^2}\right)}^{\frac{1}{D-1}}{Q}_R^{\frac{D}{D-1}}\left[1+O\left({\left(\epsilon {\left|\tau \right|}^2{Q}_R\right)}^{-\frac{2}{D-1}}\right)\right],& {Q}_R\gg \left\{\begin{array}{ll}\frac{1}{\epsilon },& \frac{1}{\epsilon {\left|\tau \right|}^2}\end{array}\right\}\end{array}\right. $$ Δ Q R , τ = BPS bound 1 + O ϵ τ 2 Q R , Q R ≪ 1 ϵ 1 ϵ τ 2 9 8 ϵ τ 2 2 + τ 2 1 D − 1 Q R D D − 1 1 + O ϵ τ 2 Q R − 2 D − 1 , Q R ≫ 1 ϵ , 1 ϵ τ 2 which leads us to the double-scaling parameter, ϵ|τ|2QR, which interpolates between the “near-BPS phase” (∆(Q) ∼ Q) and the “superfluid phase” (∆(Q) ∼ QD/(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 ∆(QR, τ) experiences a smooth transition around QR ∼ 1/|τ|2. Additionally, we find a first-order phase transition for ∆(QR, 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.