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 ∆(Q_R) for fixed and large R-charge Q_R. We focus, as an example, on the D = 3, N = 2 Wess-Zumino model with cubic superpotential W=XYZ+τ6(X3+Y3+Z3), and compute ∆(Q_{R, τ}) using the ϵ-expansion in three interesting limits. In two of these limits the (leading order) result turns out to beΔ(QR,τ)={(BPSbound)[1+O(ϵ|τ|2QR)],QR≪{1ϵ1ϵ|τ|2}98(ϵ|τ|22+|τ|2)1D−1QRDD−1[1+O((ϵ|τ|2QR)−2D−1)],QR≫{1ϵ,1ϵ|τ|2} which leads us to the double-scaling parameter, ϵ|τ|²Q_R, which interpolates between the “near-BPS phase” (∆(Q) ∼ Q) and the “superfluid phase” (∆(Q) ∼ 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 ∆(Q_{R, τ}) experiences a smooth transition around Q_R ∼ 1/|τ|². Additionally, we find a first-order phase transition for ∆(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.