On the large charge sector in the critical O(N) model at large N
Simone Giombi (Department of Physics, Princeton University, Princeton, NJ, 08544, USA); Jonah Hyman (Department of Physics, Princeton University, Princeton, NJ, 08544, USA, Department of Physics & Astronomy, University of California, Los Angeles, CA, 90095, USA)
We study operators in the rank-j totally symmetric representation of O(N) in the critical O(N) model in arbitrary dimension d, in the limit of large N and large charge j with j/N ≡ $$ \hat{j} $$ fixed. The scaling dimensions of the operators in this limit may be obtained by a semiclassical saddle point calculation. Using the standard Hubbard-Stratonovich description of the critical O(N) model at large N, we solve the relevant saddle point equation and determine the scaling dimensions as a function of d and $$ \hat{j} $$ , finding agreement with all existing results in various limits. In 4 < d < 6, we observe that the scaling dimension of the large charge operators becomes complex above a critical value of the ratio j/N, signaling an instability of the theory in that range of d. Finally, we also derive results for the correlation functions involving two “heavy” and one or two “light” operators. In particular, we determine the form of the “heavy-heavy-light” OPE coefficients as a function of the charges and d.
