On the CFT Operator Spectrum at Large Global Charge

<!--HTML--><p>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 J² &Delta;_J - ( J²/2+ J/4&nbsp;+ 3/16) &Delta;_J-1 - ( J²/2&nbsp;- J/4&nbsp;+ 3/16) &Delta;_J+1 = 0.04067 up to corrections that vanish at large J. The spectrum of low-lying excited states is also calculable explicitly: For example, the second-lowest primary operator has spin two and dimension &Delta;_J + &radic;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 (1/&radic;2)&nbsp;and &plusmn; 1/2&nbsp;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.</p>