On the CFT Operator Spectrum at Large Global Charge

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 $W={\Phi }^3$ 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 $J^2{\Delta }_J-\left(\frac{J^2}{2}+\frac{J}{4}+\frac{3}{16}\right){\Delta }_{J-1}-\left(\frac{J^2}{2}-\frac{J}{4}+\frac{3}{16}\right){\Delta }_{J+1}=0.035147$ 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 $J$.

Typos corrected. References added

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