On the CFT Operator Spectrum at Large Global Charge

About

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 = Φ3 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 ΔJ satisfies the sum rule J2 ΔJ - ( J2/2+ J/4 + 3/16) ΔJ-1 - ( J2/2 - J/4 + 3/16) ΔJ+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 ΔJ + √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 J1/2. The propagation speeds of the Goldstone waves and heavy fermions are (1/√2) and ± 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. .

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