In this note we study two-dimensional CFTs at large global charge. Since the large-charge sector decouples from the dynamics, it does not control the dynamics and an EFT 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 an NLSM description in a WKB approximation and use the large-charge sector of the solvable
SU(2)k WZW model to argue the regimes of applicability of both the large-Q expansion and the double-scaling limit.
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.
Two-dimensional conformal field theories with just Virasoro symmetry are endowed with integrable structure. We review how to construct the integrable charges in a two-dimensional conformal field theory and how to relate them to the charges of quantum sinh-Gordon theory when $c\ensuremath{\ge}25$. We then explicitly calculate the single charge characters in the large
c limit for all charges and thereby reveal how their degeneracies grow within one module. This, in particular, allows us to approximate the characters in the limit of small chemical potential, which source the respective charges. The latter gives us insights into possible transformation properties of the characters. We also comment on the full generalized Gibbs ensemble and approximations to pure states.
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.
Two-dimensional conformal field theories exhibit a universal free energy in the high temperature limit T → ∞, and a universal spectrum in the Cardy regime, Δ → ∞. We show that a much stronger form of universality holds in theories with a large central charge c and a sparse light spectrum. In these theories, the free energy is universal at all values of the temperature, and the microscopic spectrum matches the Cardy entropy for all
Δ≥6c . The same is true of three-dimensional quantum gravity; therefore our results provide simple necessary and sufficient criteria for 2d CFTs to behave holographically in terms of the leading spectrum and thermodynamics. We also discuss several applications to CFT and gravity, including operator dimension bounds derived from the modular bootstrap, universality in symmetric orbifolds, and the role of non-universal ‘enigma’ saddlepoints in the thermodynamics of 3d gravity.
Asymptotic freedom, lost: Complex conformal field theory in the two-dimensional
O(N>2) nonlinear sigma model and its realization in Heisenberg spin chains
The two-dimensional
O(N) nonlinear sigma model (NLSM) is asymptotically free for
N>2: it exhibits neither a nontrivial fixed point nor spontaneous symmetry-breaking. Here we show that a nontrivial fixed point generically does exist in the complex coupling plane and is described by a complex conformal field theory (CCFT). This CCFT fixed point is generic in the sense that it has a single relevant singlet operator, and is thus expected to arise in any non-Hermitian model with
O(N) symmetry upon tuning a single complex parameter. We confirm this prediction numerically by locating the CCFT at
N=3 in two non-Hermitian spin-1 antiferromagnetic Heisenberg chains, and in a non-Hermitian spin-
1/2 ladder, finding good agreement between the complex central charge and scaling dimensions and those obtained by analytic continuation of real fixed points from
N≤2. We further construct a realistic Lindbladian for a spin-1 chain whose no-click dynamics are governed by the non-Hermitian Hamiltonian realizing the CCFT. Since the CCFT vacuum is the eigenstate with the smallest decay rate, the system naturally relaxes under dissipative dynamics toward a CFT state, thus providing a route to preparing long-range entangled states through engineered dissipation.