  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 $W = \Phi^3$ 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 $\Delta_J$ satisfies the sum rule $
J^2 \Delta_J - \left( \tfrac{J^2}{2} + \tfrac{J}{4} + \tfrac{3}{16} \right)
\Delta_{J-1} - \left( \tfrac{J^2}{2} - \tfrac{J}{4} + \tfrac{3}{16} \right)
\Delta_{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 $\Delta\ll J +
\sqrt{3}$. 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 $J$.
