title: Emergent descriptions at large charge: A foray into the structure of conformal field theories and beyond
Creators: name: Moser, Rafael
Publisher: name: Universität Bern
Year: 2023
Subjects: subject: 530 Physics

description: Conformal Field Theories (CFT)s play a central role in the study of Quantum Field Theory (QFT). They represent the fixed point of the Wilsonian Renormalization Group (RG) flow and any QFT is in principle describable as a relevant deformation of the associated nearby Conformal Field Theory (CFT). 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 Spontaneous Symmetry Breaking (SSB). We discuss the symmetry-constraints imposed by conformal invariance on the quantum theory, introduce the concept of CFT data and the Operator–Product Expansion (OPE). Concerning SSB, we discuss the existence of Nambu–Goldstone (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 present the LCE in the broader context of the different methods available for accessing CFT data. Particularly, we discuss its relation to large-spin expansions in CFTs and the description of operators with both large spin and large charge. We discuss the emergence of effective condensed-matter descriptions, in particular superfluids, in correlators involving states with large global quantum numbers. Finally, we use the superfluid Effective Field Theory (EFT) description to systematically study two-, three- and four-point functions for CFTs with a global O(2) symmetry. Using the EFT approach we derive universal results for the spectrum of scaling dimensions and three-point coefficients at large charge.
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 = 3Wilson–Fisher (WF) fixed point at large N and derive the leading order asymptotics at large charge Q in the double scaling limit Q/N fixed, where scaling dimensions can be studied analytically in the limit Q/2N ≫1, where we recover the superfluid EFT structure, and Q/2N ≪1, where we recover the free mean-field limit. These limits can be connected by resurgent analysis. We also study the spectrum of fluctuations to confirm EFT predictions. Next, we use a fixed-charge approach to gain access to the leading order effective potential for the ϕ4 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. In D = 5, under the assumption of unitarity the ϕ4-model does not appear to be Ultra–Violet (UV) complete. Finally, we discuss the interacting fixed points of three-dimensional fermionic CFTs in the double-scaling limit of large charge and large N. While the Gross–Neveu (GN)model exhibits a Fermi-sphere description at large charge, whose fate at finite N is yet to be determined, for the Nambu–Jona–Lasinio (NJL)-type models we find a Bose–Einstein Condensate (BEC). The large-charge sector of these models is therefore captured by the superfluid EFT approach.