On the CFT Operator Spectrum at Large Global Charge

TL;DR

This work shows that the large global charge sector of certain 3D CFTs is governed by a conformal Goldstone effective action for the phase of a complex order parameter, enabling a controlled $1/J$ expansion of operator dimensions. In both the bosonic $O(2)$ model and the SUSY fixed point with $W=\Phi^3$, the ground-state dimension scales as $\Delta_J \sim c_{3/2}J^{3/2}$ with a universal $J^0$ term $-0.0937256$ arising from one-loop Casimir energy, and excited states correspond to free oscillator modes on $S^2$ with specific spin content. Fermions decouple in the large-$J$ limit, leaving the same Goldstone universality class as the bosonic model, with distinct but compatible dispersion relations dictated by superconformal symmetry. The framework extends to higher dimensions and dual descriptions (monopole/S-duality) and connects to Weyl anomaly data and bootstrap approaches, suggesting a robust, broadly applicable picture of large-charge spectra in strongly coupled CFTs.

Abstract

We calculate the anomalous dimensions of operators with large global charge $J$ in certain strongly coupled conformal field theories in three dimensions, such as the O(2) model and the supersymmetric fixed point with a single chiral superfield and a $W = Φ^3$ superpotential. Working in a $1/J$ expansion, we find that the large-$J$ sector of both examples is controlled by a conformally invariant effective Lagrangian for a Goldstone boson of the global symmetry. For both these theories, we find that the lowest state with charge $J$ is always a scalar operator whose dimension $Δ_J$ satisfies the sum rule $ J^2 Δ_J - \left( \tfrac{J^2}{2} + \tfrac{J}{4} + \tfrac{3}{16} \right) Δ_{J-1} - \left( \tfrac{J^2}{2} - \tfrac{J}{4} + \tfrac{3}{16} \right) Δ_{J+1} = 0.035147 $ up to corrections that vanish at large $J$. The spectrum of low-lying excited states is also calculable explcitly: For example, the second-lowest primary operator has spin two and dimension $Δ\ll J + \sqrt{3}$. In the supersymmetric case, the dimensions of all half-integer-spin operators lie above the dimensions of the integer-spin operators by a gap of order $J^{1/2}$. The propagation speeds of the Goldstone waves and heavy fermions are $\frac{1}{\sqrt{2}}$ and $\pm \frac{1}{2}$ times the speed of light, respectively. These values, including the negative one, are necessary for the consistent realization of the superconformal symmetry at large $J$.