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arXiv:2207.08919 · Journal of High Energy Physics

Identifying Large Charge operators

Gil Badel, Alexander Monin, Riccardo Rattazzi A

Published Feb 13, 2023• 4 citations

Abstract

Abstract 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.

DOI: 10.1007/jhep02(2023)119

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A bstract We consider correlators for the flux of energy and charge in the background of operators with large global U(1) charge in conformal field theory (CFT). It has recently been shown that the corresponding Euclidean correlators generically admit a semiclassical description in terms of the effective field theory (EFT) for a conformal superfluid. We adapt the semiclassical description to Lorentzian observables and compute the leading large charge behavior of the flux correlators in general U(1) symmetric CFTs. We discuss the regime of validity of the large charge EFT for these Lorentzian observables and the subtleties in extending the EFT approach to subleading corrections. We also consider the Wilson-Fisher fixed point in d = 4 − ϵ dimensions, which offers a specific weakly coupled realization of the general setup, where the subleading corrections can be systematically computed without relying on an EFT.

Abstract We study operators in Schrödinger invariant field theories (non-relativistic conformal field theories or NRCFTs) with large charge (particle number) and spin. Via the state-operator correspondence for NRCFTs, such operators correspond to states of a superfluid in a harmonic trap with phonons or vortices. Using the effective field theory of the Goldstone mode, we compute the dimensions of operators to leading order in the angular momentumLand chargeQ. We find a diverse set of scaling behaviors for NRCFTs in bothd= 2 andd= 3 spatial dimensions. These results apply to theories with a superfluid phase, such as unitary fermions or critical anyon systems.

Hellerman et al. (arXiv:1505.01537) have shown that in a generic CFT the spectrum of operators carrying a large U(1) charge can be analyzed semiclassically in an expansion in inverse powers of the charge. The key is the operator state correspondence by which such operators are associated with a finite density superfluid phase for the theory quantized on the cylinder. The dynamics is dominated by the corresponding Goldstone hydrodynamic mode and the derivative expansion coincides with the inverse charge expansion. We illustrate and further clarify this situation by first considering simple quantum mechanical analogues. We then systematize the approach by employing the coset construction for non-linearly realized space-time symmetries. Focussing on CFT3 we illustrate the case of higher rank and non-abelian groups and the computation of higher point functions. Three point function coefficients turn out to satisfy universal scaling laws and correlations as the charge and spin are varied.

This thesis explores the application of semiclassical methods in the study of states with large quantum numbers for theories invariant under internal symmetries. In the first part of the thesis, we study zero-temperature superfluids. These provide a general description of many systems at finite charge density. In particular, we derive a universal effective field theory description for non-Abelian superfluids. Such construction illustrates the role of gapped Goldstones, Goldstone modes whose gap is fixed by the symmetry, and may be as large as the strong coupling scale of the system. The second and third part of the thesis are devoted to the study of operators with large internal charge in strongly coupled conformal field theories. Using effective field theory techniques, we derive universal results for the spectrum of scaling dimensions and the OPE coefficients in a large charge expansion, both for theories invariant under Abelian and non-Abelian symmetry groups. We also extend these results to operators with large spin as well as large internal charge. The last part of this thesis studies operators with large internal charge within the ε-expansion. We show how, using a semiclassical approach, one can overcome the breakdown of diagrammatic perturbation theory for the multi-legged amplitudes associated with these operators. These results provide a concrete illustration of the systematic large charge expansion discussed in the previous parts.

We show that the correlator of three large charge operators with minimal scaling dimension can be computed semiclassically in conformal field theories (CFTs) with a

U(1)

symmetry for arbitrary fixed values of the ratios of their charges. We obtain explicitly the operator product expansion coefficient from the numerical solution of a nonlinear boundary value problem in the conformal superfluid effective field theory in

3d

. The result applies in all three-dimensional CFTs with a

U(1)

symmetry whose large charge sector is a superfluid.