Working in sectors of large global charge leads to important simplifications when studying strongly coupled CFTs. In this talk I will introduce the large-charge expansion via the simple example of the O(2) model

The large charge approach allows to describe special sectors of strongly-coupled systems. When another controlling parameter is present, it is still useful because it allows to unveil interesting mathematical structures and access previously unexplored parts of the spectrum. In this talk I will concentrate on the O(N) vector model in the large-N limit, give a geometrical interpretation of the asymptotic expansion of the dimension of the lowest operator of fixed charge, and use resurgence to resum this expansion. The result suggests an unexpected interplay between conformal symmetry and resurgence and will lead to an intuitive interpretation for the impressive agreement of the large-charge predictions with lattice data also in the O(2) model case. Based on 1610.04495, 2008.03308, 2102.12488.

The large charge approach allows to describe special sectors of strongly-coupled systems. When another controlling parameter is present, it is still useful because it allows to unveil interesting mathematical structures and access previously unexplored parts of the spectrum. In this talk I will concentrate on the O(N) vector model in the large-N limit, give a geometrical interpretation of the asymptotic expansion of the dimension of the lowest operator of fixed charge, and use resurgence to resum this expansion. The result suggests an unexpected interplay between conformal symmetry and resurgence and will lead to an intuitive interpretation for the impressive agreement of the large-charge predictions with lattice data also in the O(2) model case. Based on 1610.04495, 2008.03308, 2102.12488.

The large-charge master field which generates all n-point correlation functions with an insertion of large charge Q in non-relativistic conformal field theory is obtained. This field is used to compute Schr"odinger-invariant n-point correlation functions of large-charge operators via a direct evaluation of the path integral. Conformal dimensions are found to agree with calculations based on the state-operator correspondence. The master field solution exhibits an emergent harmonic trap whose frequency is a function of the Euclidean time. The large-charge effective action with operator insertions describes a droplet of superfluid matter whose spatial size scales with the time separation of sources. The solution is used to compute Schrödinger symmetry breaking corrections in the large-charge effective field theory (EFT) due to a finite scattering length in the fundamental theory of fermions near unitarity. The scaling of these effects in the large-charge power counting scheme is established, and the size of the effects is quantified using input from quantum Monte Carlo simulations of the near-unitary gas, as well as from the large-N expansion at large charge.

We study the unitary Fermi gas in a harmonic trapping potential starting from a microscopic theory in the limit of large charge and large number of fermion flavors N. In this regime, we present an algorithmic procedure for extracting data from perturbation theory, order-by-order, without the need for other assumptions. We perform a gradient expansion in the interior of the particle cloud, sufficiently far from the cloud edge where the particle density drops rapidly to zero. In this latter region we present the first microscopic computation characterizing the contribution of the edge terms. The microscopic theory reproduces the predictions of the superfluid EFT, including the action, the form of the gap equation, and the energy of the system in a harmonic trap (which maps, via the non-relativistic state-operator correspondence, to the scaling dimension of the lowest operator of charge Q). We additionally give the Wilsonian coefficients at leading order in N up to NNLO in the large-charge expansion.

We study the spectrum of scalar charged operators in CFTs with a U(1) global symmetry. The charged operators are dual, by the state-operator correspondence, to homogenous charged states on the sphere. Such states can break the U(1) symmetry, and we define what we call the large ff regime in the CFT as one where the symmetry breaking scale is much higher than the scale of the CFT sphere. In such a regime, there is (an approximate) Goldstone boson associated to the breaking. We show that consistency of the Goldstone boson physics implies that the spectrum of states, and therefore of operators, must be convex in charge. More precisely, we show that any family of operators of different charges, which are lowest dimension of their charge, and which additionally share the same realisation of the Goldstone boson in terms of the degrees of freedom of the CFT, must be convex.