We extend and clarify the large-charge expansion of the conformal dimension ${\mathrm{\ensuremath{\Delta}}}_{Q}$ of the lowest operator of charge
Q in nonrelativistic conformal field theories using the state-operator correspondence. The latter requires coupling the theory to an external harmonic trap that confines the particles to a spherical cloud, at the edge of which the effective theory breaks down and leads to divergences. Only recently has this issue been overcome by constructing appropriate counterterms at the edge of the cloud [S. Hellerman and I. Swanson, arXiv:2010.07967]. In this paper, we extend these results by systematically analyzing the degree of divergence of operators in the effective action and show that there always exist appropriate edge counterterms that make the final contributions to ${\mathrm{\ensuremath{\Delta}}}_{Q}$ finite. On the other side of the correspondence, this also provides new corrections to the Thomas-Fermi approximation of the unitary Fermi gas, and we comment on their relevance for ultracold atom physics.
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
We calculate the contributions of a general non-vacuum conformal family to Rényi entropy in two-dimensional conformal field theory (CFT). The primary operator of the conformal family can be either non-chiral or chiral, and we denote its scaling dimension by Δ . For the case of two short intervals on a complex plane, we expand the Rényi mutual information by the cross ratio x to order x 2 Δ +2 . For the case of one interval on a torus with low temperature, we expand the Rényi entropy by q =exp(−2π β /L), with β being the inverse temperature and L being the spatial period, to order q Δ +2 . To make the result meaningful, we require that the scaling dimension Δ cannot be too small. For two intervals on a complex plane we need Δ > 1, and for one interval on a torus we need Δ > 2. We work in the small Newton constant limit on the gravity side and so a large central charge limit on the CFT side, and find matches of gravity and CFT results.
We consider the large-charge expansion of the charged ground state of a Schrodinger-invariant, nonrelativistic conformal field theory in a harmonic trap, in general dimension d. In the existing literature, the energy in the trap has been computed to next-to-leading order (NLO) at large charge Q, which comes from the classical contribution of two higher-derivative terms in the effective field theory. In this note, we explain the structure of operators localized at the edge of the droplet, where the density drops to zero. We list all operators contributing to the ground-state energy with nonnegative powers of Q in the large-Q expansion. As a test, we use dimensional regularization to reproduce the calculation of the NLO ground state energy by Kravec and Pal , and we recover the same universal coefficient for the logarithmic term as in that work. We refine the derivation by presenting a systematic operator analysis of the possible edge counterterms, showing that different choices of cutoff procedures must yield the same renormalized result up to an enumerable list of Wilson coefficients for conformally invariant local counterterms at the droplet edge. We also demonstrate the existence of a previously unnoticed edge contribution to the ground-state operator dimension of order Q^{{2\over 3} - {1\over d}} in d spatial dimensions. Finally, we show there is no bulk or edge counterterm scaling as Q^0 in two spatial dimensions, which establishes the universality of the order Q^0 term in large-Q expansion of the lowest charged operator dimension in d=2.
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.
We analyze the operator spectrum of Conformal Field Theories at large conformal dimensions
Delta with and without global symmetries. General constraints of crossing symmetry and unitarity allow us to extract a number of universal properties of heavy operator spectrum perturbatively in
1overDelta.\nFirst, we consider four-point correlators and solve crossing equations in the deep Euclidean regime. Large scaling dimension
Delta tails of the weighted spectral density of primary operators of given spin in one channel are matched to the Euclidean OPE data in the other channel. Subleading
1overDelta tails are systematically captured by including more operators in the Euclidean OPE in the dual channel. We use dispersion relations for conformal partial waves in the complex
Delta plane, the Lorentzian inversion formula and complex tauberian theorems to derive this result. We make predictions for the 3d Ising model.\nSecond, we apply the methods of tauberian theory to modular invariance in 2d CFTs. We derive lower and upper bounds on the number of operators within a given energy interval. At high energies we rigorously derive the Cardy formula for the microcanonical entropy together with optimal error estimates for various widths of the averaging energy shell. We identify a new universal contribution to the microcanonical entropy controlled by the central charge and the width of the shell. We derive an upper bound on the spacings between Virasoro primaries. Analogous results are obtained in holographic 2d CFTs. \nThird, we consider unitary CFTs with continuous global symmetries in
d>2. We consider a state created by the lightest operator of large charge
Qgg1 and analyze the correlator of two light charged operators in this state. Assuming the correlator admits a well-defined large
Q expansion and, relatedly, that the macroscopic (thermodynamic) limit of the correlator exists, we find that the crossing equations admit a consistent truncation, where only a finite number
N of Regge trajectories contribute at leading nontrivial order. We classify all such solutions to the crossing. For one Regge trajectory
N=1, the solution is unique and given by the effective field theory of a Goldstone mode.