On the large charge sector in the critical O(N) model at large N

Simone Giombi,Jonah Hyman

doi:10.1007/jhep09(2021)184

Abstract

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.

Highlights

This leads to a set of Feynman diagrammatic rules where one uses an induced σ propagator and the σφiφi vertex. This standard 1/N perturbation theory works as long as one considers correlation functions of operators with quantum numbers that are finite in the large N limit
To obtain the scaling dimension, we study directly the two-point function of the large charge operators on Rd, and determine the semiclassical saddle point for the σ field as a function of the insertion points of the “heavy” operators
In this paper we have studied large charge operators in the large N critical O(N ) model in general d, in the limit where the charge j goes to infinity withj = j/N fixed

Summary

The saddle point equation

Let us consider a scalar composite operator Oj in the spin j totally symmetric traceless representation of O(N ). The key observation is that we may view σ∗ as the one-point function of σ in the presence of the large charge operators. Recalling that σ in the critical O(N ) model is an operator of scaling dimension ∆ = 2 + O(1/N ), and using the form of the three-point function of scalar operators fixed by conformal symmetry. Where cσ is an undetermined constant that should be fixed by solving the saddle point equation.3 This is obtained by extremizing the effective action in (2.5), and reads δ δσ(x). In order to solve for the constant cσ in (2.8), we will need to evaluate explicitly the Green’s function G(x, y; σ∗) This is a non-trivial calculation, which we carry out in the subsection

The Green’s function The Green’s function is the solution to

The functional determinant

The scaling dimension

Correlation functions at large charge

Three-point functions

Four-point functions

Conclusion