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arXiv:2309.14428 · Journal of High Energy Physics

Flux correlators and semiclassics

Eren Firat, Alexander Monin, Riccardo Rattazzi A, Matthew T. Walters

Published Mar 12, 2024• 7 citations
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Abstract

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.

DOI
10.1007/jhep03(2024)067

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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 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.
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.
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.
This thesis aims to explore the structure of CFTs with global internal symmetries and beyond via the Large-Charge Expansion (LCE), a semi-classical expansion applicable for states with large global quantum numbers. In the first part of this thesis we study CFT and SSB. We discuss the symmetry-constraints imposed by conformal invariance, introduce the concept of CFT data and the OPE. Concerning SSB, we discuss the existence of NG modes, the general counting rule for the number of NG modes under the spontaneous breaking of global internal symmetries and a generalization of the Goldstone theorem at finite density. In the second part of this thesis we discuss the current state-of-the-art understanding of the LCE and systematically study CFTs with a global O(2) symmetry in the context of the LCE. We discuss the emergence of effective condensed-matter descriptions, in particular superfluids, in correlators involving states with large global quantum numbers. We then use the superfluid EFT description to systematically study two-, three- and four-point functions for CFTs with a global O(2) symmetry. In the last part of this thesis we study CFTs in the double-scaling limit of large charge and large N . We discuss the D = 3 WF fixed point at large N and derive the leading order asymptotics at large charge Q in the double scaling limit Q/N fixed. Next, we use a fixed-charge approach to gain access to the leading order effective potential for the quartic theory, which we then study for spacetime dimensions 2 < D < 6. In D = 3, we reproduce and extend old results originally found by re-summing Feynman diagrams. Finally, we discuss the interacting fixed points of three-dimensional fermionic CFTs in the double-scaling limit. While the GN model exhibits a Fermi-sphere description at large charge, whose fate at finite N is yet to be determined, for the NJL-type models we find a BEC.
The large-charge master field which generates all n -point correlation functions with an insertion of large charge Q in nonrelativistic conformal field theory is obtained. This field is used to compute Schrödinger-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. Published by the American Physical Society 2024