Large Global Charge Research Articles

Spinning correlators in large-charge CFTs

We systematically study correlators of a generic conformal field theory with a global O(2) symmetry in a sector of large global charge. We focus in particular on three- and four-point correlators with conserved current insertions sandwiched between spinful excited states corresponding to phonons over the large-charge vacuum. We also discuss loop corrections to the scaling dimensions and observe the presence of multiple logarithms in even dimensions.

2D CFTs: Large charge is not enough to control the dynamics

In this note we study two-dimensional conformal field theories at large global charge. Since the large-charge sector decouples from the dynamics, it does not control the dynamics and an effective field theory construction that works in higher-dimensional theories fails. It is however possible to use large charge in a double-scaling limit when another controlling parameter is present. We find some general features of the spectrum of models that admit a nonlinear sigma model description in a Wentzel-Kramers-Brillouin approximation and use the large-charge sector of the solvable $SU(2{)}_{k}$ Wess-Zumino-Witten model to argue the regimes of applicability of both the large-$Q$ expansion and the double-scaling limit.

Observables in inhomogeneous ground states at large global charge

As a sequel to previous work, we extend the study of the ground state configuration of the D = 3, Wilson-Fisher conformal O(4) model. In this work, we prove that for generic ratios of two charge densities, ρ1/ρ2, the ground-state configuration is inhomogeneous and that the inhomogeneity expresses itself towards longer spatial periods. This is the direct extension of the similar statements we previously made for ρ1/ρ2 ≪ 1. We also compute, at fixed set of charges, ρ1, ρ2, the ground state energy and the two-point function(s) associated with this inhomogeneous configuration on the torus. The ground state energy was found to scale (ρ1 + ρ2)3/2, as dictated by dimensional analysis and similarly to the case of the O(2) model. Unlike the case of the O(2) model, the ground also strongly violates cluster decomposition in the large-volume, fixed-density limit, with a two-point function that is negative definite at antipodal points of the torus at leading order at large charge.

Accessing large global charge via the \u03f5-expansion

We compute the lowest operator dimension ∆(J; D) at large global charge J in the O(2) Wilson-Fisher model in D = 4 − ϵ dimensions, to leading order in both 1/J and ϵ. While the effective field theory approach of [1] could only determine ∆(J; 3) as a series expansion in 1/J up to an undetermined constant in front of each term, this time we try to determine the coefficient in front of J3/2 in the ϵ-expansion. The final result for ∆(J; D) in the (resummed) ϵ-expansion, valid when J ≫ 1/ϵ ≫ 1, turns out to beΔJD=2D−13D−29D−2π5DD2D−15ΓD224π21D−1ϵD−12D−1×JDD−1+OJD−2D−1\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\Delta \\left(J;D\\right)=\\left[\\frac{2\\left(D-1\\right)}{3\\left(D-2\\right)}{\\left(\\frac{9\\left(D-2\\right)\\pi }{5D}\\right)}^{\\frac{D}{2\\left(D-1\\right)}}{\\left[\\frac{5\\Gamma \\left(\\frac{D}{2}\\right)}{24{\\pi}^2}\\right]}^{\\frac{1}{D-1}}{\\epsilon}^{\\frac{D-1}{2\\left(D-1\\right)}}\\right]\\times {J}^{\\frac{D}{D-1}}+O\\left({J}^{\\frac{D-2}{D-1}}\\right) $$\\end{document}where next-to-leading order onwards were not computed here due to technical cumbersomeness, despite there are no fundamental difficulties. We also compare the result at ϵ = 1,ΔJ=0.293×J3/2+⋯\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\Delta (J)=0.293\\times {J}^{3/2}+\\cdots $$\\end{document}to the actual data from the Monte-Carlo simulation in three dimensions [2], and the discrepancy of the coefficient 0.293 from the numerics turned out to be 13%. Additionally, we also find a crossover of ∆(J; D) from ∆(J) ∝ {J}^{frac{D}{D-1}} to ∆(J) ∝ J, at around J ∼ 1/ϵ, as one decreases J while fixing ϵ (or vice versa), reflecting the fact that there are no interacting fixed-point at ϵ = 0. Based on this behaviour, we propose an interesting double-scaling limit which fixes λ ≡ Jϵ, suitable for probing the region of the crossover. I will give ∆(J; D) to next-to-leading order in perturbation theory, either in 1/λ or in λ, valid when λ ≫ 1 and λ ≪ 1, respectively.

Superfluids, vortices and spinning charged operators in 4d CFT

We include vortices in the superfluid EFT for four dimensional CFTs at large global charge. Using the state-operator correspondence, vortices are mapped to charged operators with large spin and we compute their scaling dimensions. Different regimes are identified: phonons, vortex rings, Kelvin waves, and vortex crystals. We also compute correlators with a Noether current insertion in between vortex states. Results for the scaling dimensions of traceless symmetric operators are given in arbitrary spacetime dimensions.

A note on inhomogeneous ground states at large global charge

In this note we search for the ground state of the D = 3 Wilson-Fisher conformal O(4) model, at nonzero values of the two independent charge densities ρ1,2, on the torus spatial slice. Using an effective theory valid on scales longer than the scale defined by the charge density, we show that the ground-state configuration is inhomogeneous for generic ratios ρ1/ρ2. This result confirms, within the context of a well-defined effective theory, a recent no-go result of [1]. We also show that any spatially periodic ground state solutions have an energetic preference towards longer periods, within some range of ρ1/ρ2 containing a neighborhood of zero. This suggests that the scale of variation of the ground state solution in finite volume will be the infrared scale, and that the use of the effective theory at large charge in finite volume is self-consistent. Note added: the statements in this paper are true for arbitrary ratio of ρ1/ρ2, which we proved after we uploaded this paper. See [2].

An AdS/EFT correspondence at large charge

Considering theories in sectors of large global charge Q results in a semiclassical effective field theory (eft) description for some strongly-coupled conformal field theories (cfts) with continuous global symmetries. Hence, when studying dualities at large charge, we can have control over the strongly coupled side of the duality and gain perturbative access to both dual pairs.In this work we discuss the AdS/CFT correspondence in the regime Q≫CT≫1 where both the eft and gravity descriptions are valid and stable (CT being the central charge). We present the observation that the ground state energy as a function of the Abelian charge Q for a simple eft in some three-dimensional cfts coincides with the expression for the mass of an anti-de Sitter–Reissner–Nordström black hole as a function of its charge. Using this observation we propose a tentative dictionary relating cft, eft and holographic descriptions. We also find agreement for the higher-derivative corrections on both sides, suggesting a large-CT expansion on the eft side.

A matrix CFT at multiple large charges

We investigate matrix models in three dimensions where the global SU(N ) symmetry acts via the adjoint map. Analyzing their ground state which is homogeneous in space and can carry either a unique or multiple fixed charges, we show the existence of at least two distinct fixed points of the renormalization group (rg) flow. In particular, the one type of those fixed points manifests itself via tractable deviations in the large-charge expansion from the known predictions in the literature. We demonstrate most of the novel features using mainly the example of the SU(4) matrix theory to compute the anomalous dimension of the lowest scalar operator with large global charge(s).

Rotating superfluids and spinning charged operators in conformal field theory

We calculate the scaling dimensions of operators with large global charge and spin in 2+1 dimensional conformal field theories. By the state-operator correspondence, these operators correspond to superfluids with vortices and can be systematically studied using effective field theory. As the spin increases from zero to the unitarity bound, the superfluid state corresponding to the lowest dimension operator passes through three distinct regimes: (1) a single phonon, (2) two vortices, and (3) multiple vortices. We also calculate correlation functions with two such operators and the Noether current.

Abelian scalar theory at large global charge

AbstractWe elaborate on Abelian complex scalar models, which are dictated by natural actions (all couplings are of order one), at fixed and large global U(1) charge in an arbitrary number of dimensions. The ground state is coherently constructed by the zero modes and the appearance of a centrifugal potential is quantum mechanically verified. Using the path integral formulation we systematically analyze the quantum fluctuations around in order to derive an effective action for the Goldstone mode, which becomes perturbatively meaningful when the charge is large. In this regime we explicitly show, by computing the first few loop corrections, that the whole construction is stable against quantum effects, in the sense that any higher derivative couplings to Goldstone's tree‐level action are suppressed by appropriate powers of the large charge.

On the CFT operator spectrum at large global charge

We calculate the anomalous dimensions of operators with large global charge $J$ in certain strongly coupled conformal field theories in three dimensions, such as the O(2) model and the supersymmetric fixed point with a single chiral superfield and a $W = \Phi^3$ superpotential. Working in a $1/J$ expansion, we find that the large-$J$ sector of both examples is controlled by a conformally invariant effective Lagrangian for a Goldstone boson of the global symmetry. For both these theories, we find that the lowest state with charge $J$ is always a scalar operator whose dimension $\Delta_J$ satisfies the sum rule $ J^2 \Delta_J - \left( \tfrac{J^2}{2} + \tfrac{J}{4} + \tfrac{3}{16} \right) \Delta_{J-1} - \left( \tfrac{J^2}{2} - \tfrac{J}{4} + \tfrac{3}{16} \right) \Delta_{J+1} = 0.035147 $ up to corrections that vanish at large $J$. The spectrum of low-lying excited states is also calculable explcitly: For example, the second-lowest primary operator has spin two and dimension $\Delta\ll J + \sqrt{3}$. In the supersymmetric case, the dimensions of all half-integer-spin operators lie above the dimensions of the integer-spin operators by a gap of order $J^{1/2}$. The propagation speeds of the Goldstone waves and heavy fermions are $\frac{1}{\sqrt{2}}$ and $\pm \frac{1}{2}$ times the speed of light, respectively. These values, including the negative one, are necessary for the consistent realization of the superconformal symmetry at large $J$.

On the universality class of certain string theory hadrons

Exploiting the gauge/gravity correspondence we find the spectrum of hadronic-like bound states of adjoint particles with a large global charge in several confining theories. In particular, we consider an embedding of four-dimensional N = 1 supersymmetric Yang–Mills into IIA string theory, two classes of three-dimensional gauge theories and the softly broken version of one of them. In all cases we describe the low energy excitations of a heavy hadron with mass proportional to its global charge. These excitations include: the hadron's nonrelativistic motion, its stringy excitations and excitations corresponding to the addition of massive constituents. Our analysis provides ample evidence for the universality of such hadronic states in confining theories admitting supergravity duals. Besides, we find numerically a new smooth solution that can be thought of as a nonsupersymmetric deformation of G 2 holonomy manifolds.

Analytic non-supersymmtric background dual of a confining gauge theory and the corresponding plane wave theory of Hadrons

We find a regular analytic 1st order deformation of the Klebanov-Strassler background. From the dual gauge theory point of view the deformation describes supersymmetry soft breaking gaugino mass terms. We calculate the difference in vacuum energies between the supersymmetric and the non-supersymmetric solutions and find that it matches the field theory prediction. We also discuss the breaking of the $U(1)_R$ symmetry and the space-time dependence of the gaugino bilinears two point function. Finally, we determine the Penrose limit of the non-supersymmetric background and write down the corresponding plane wave string theory. This string describes ``annulons''-heavy hadrons with mass proportional to large global charge. Surprisingly the string spectrum has two fermionic zero modes. This implies that the sector in the non-supersymmetric gauge theory which is the dual of the annulons is supersymmetric.

A soluble string theory of hadrons

We consider Penrose limits of the Klebanov-Strassler and Maldacena-Nunez holographic duals to N =1 supersymmetric Yang-Mills. By focusing in on the IR region we obtain exactly solvable string theory models. These represent the nonrelativistic motion and low-lying excitations of heavy hadrons with mass proportional to a large global charge. We argue that these hadrons, both physically and mathematically, take the form of heavy nonrelativistic strings; we term them "annulons." A simple toy model of a string boosted along a compact circle allows us considerable insight into their properties. We also calculate the Wilson loop carrying large global charge and show the effect of confinement is quadratic, not linear, in the string tension.