We study the scaling dimension ∆ φ n of the operator φ n where φ is the fundamental complex field of the U (1) model at the Wilson-Fisher fixed point in d = 4 − ε. Even for a perturbatively small fixed point coupling λ * , standard perturbation theory breaks down for sufficiently large λ * n. Treating λ * n as fixed for small λ * we show that ∆ φ n can be successfully computed through a semiclassical expansion around a non-trivial trajectory, resulting in We explicitly compute the first two orders in the expansion, ∆ −1 (λ * n) and ∆ 0 (λ * n). The result, when expanded at small λ * n, perfectly agrees with all available diagrammatic computations. The asymptotic at large λ * n reproduces instead the systematic large charge expansion, recently derived in CFT. Comparison with Monte Carlo simulations in d = 3 is compatible with the obvious limitations of taking ε = 1, but encouraging.

In arXiv:1909.01269 it was shown that the scaling dimension of the lightest charge n operator in the U (1) model at the Wilson-Fisher fixed point in D = 4 − ε can be computed semiclassically for arbitrary values of λn, where λ is the perturbatively small fixed point coupling. Here we generalize this result to the fixed point of the U (1) model in 3 − ε dimensions. The result interpolates continuously between diagrammatic calculations and the universal conformal superfluid regime for CFTs at large charge. In particular it reproduces the expectedly universal O(n 0 ) contribution to the scaling dimension of large charge operators in 3D CFTs. * 1 The claims made in a recent revival [5] seem controversial. See indeed [6] for a critical perspective. 2 A similar double expansion exists at large N [14].

The Large Charge sector of Conformal Field Theory (CFT) can generically be described through a semiclassical expansion around a superfluid background. In this work, focussing on U (1) invariant Wilson-Fisher fixed points, we study the spectrum of spinning large charge operators. For sufficiently low spin these correspond to the phonon excitations of the superfluid state. We discuss the organization of these states into conformal multiplets and the form of the corresponding composite operators in the free field theory limit. The latter entails a mapping, built order-by-order in the inverse charge n −1 , between the Fock space of vacuum fluctuations and the Fock space of fluctuations around the superfluid state. We discuss the limitations of the semiclassical method, and find that the phonon description breaks down for spins of order n 1/2 while the computation of observables is valid up to spins of order n. Finally, we apply the semiclassical method to compute some conformal 3-point and 4-point functions, and analyze the conformal block decomposition of the latter with our knowledge of the operator spectrum.

The Large Charge sector of Conformal Field Theory (CFT) can generically be described through a semiclassical expansion around a superfluid background. In this work, focussing on U(1) invariant Wilson-Fisher fixed points, we study the spectrum of spinning large charge operators. For sufficiently low spin these correspond to the phonon excitations of the superfluid state. We discuss the organization of these states into conformal multiplets and the form of the corresponding composite operators in the free field theory limit. The latter entails a mapping, built order-by-order in the inverse charge n−1, between the Fock space of vacuum fluctuations and the Fock space of fluctuations around the superfluid state. We discuss the limitations of the semiclassical method, and find that the phonon description breaks down for spins of order n1/2 while the computation of observables is valid up to spins of order n. Finally, we apply the semiclassical method to compute some conformal 3-point and 4-point functions, and analyze the conformal block decomposition of the latter with our knowledge of the operator spectrum.