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arXiv:2103.01331 · Physical review. D/Physical review. D.

OPE meets semiclassics

Gabriel Cuomo

Published Apr 16, 2021• 18 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevd.103.085005

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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)U(1)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 3d3d3d. The result applies in all three-dimensional CFTs with a U(1)U(1)U(1) symmetry whose large charge sector is a superfluid.
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.
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.
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.
This thesis studies the asymptotic spectrum of three-dimensional conformal field theories with a global U(1) symmetry. Using a thermal effective action on S1 x S2, with sources for rotation and charge, we derive the coarse-grained density of states at large scaling dimension, spin, and charge. Conformal characters are used to distinguish fixed angular momentum component along a fixed direction from fixed total spin and to remove descendants, yielding an explicit asymptotic density of primary operators. At leading order, the result depends on the thermal free-energy coefficient and the charge susceptibility; charge produces a Gaussian suppression whose form is determined by an inverse Laplace–Fourier transform. We establish the regime in which the high-temperature, small-holonomy, and saddle-point approximations are controlled. A formal continuation towards the spectral edge reproduces the expected large-charge scaling Delta ~ q^3/2 and a small-spin correction J^2/q^3/2, but cannot determine their coefficients reliably because the relevant holonomy is of order one. The framework is tested in free scalar and fermion theories and against Kerr–Newman–AdS thermodynamics through the AdS/CFT correspondence.
We explore the consequences of conformal symmetry for the operator product expansions in nonrelativistic field theories. Similar to the relativistic case, the OPE coefficients of descendants are related to that of the primary. However, unlike relativistic CFTs the 3-point function of primaries is not completely specified by conformal symmetry. Here, we show that the 3-point function between operators with nonzero particle number, where (at least) one operator has the lowest dimension allowed by unitarity, is determined up to a numerical coefficient. We also look at the structure of the family tree of primaries with zero particle number and discuss the presence of conservation laws in this sector.
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.