Semiclassics, Goldstone Bosons and CFT data (1611.02912v1)

Abstract: Hellerman et al. (arXiv:1505.01537) have shown that in a generic CFT the spectrum of operators carrying a large U(1) charge can be analyzed semiclassically in an expansion in inverse powers of the charge. The key is the operator state correspondence by which such operators are associated with a finite density superfluid phase for the theory quantized on the cylinder. The dynamics is dominated by the corresponding Goldstone hydrodynamic mode and the derivative expansion coincides with the inverse charge expansion. We illustrate and further clarify this situation by first considering simple quantum mechanical analogues. We then systematize the approach by employing the coset construction for non-linearly realized space-time symmetries. Focussing on CFT3 we illustrate the case of higher rank and non-abelian groups and the computation of higher point functions. Three point function coefficients turn out to satisfy universal scaling laws and correlations as the charge and spin are varied.

• The paper applies semiclassical methods to analyze the spectrum of large charge operators in CFTs.
• It develops effective field theories via coset construction to capture Goldstone boson dynamics arising from symmetry breaking.
• It extends these techniques to non-Abelian symmetries, offering insights into operator scaling dimensions and n-point correlation functions.

Semiclassics, Goldstone Bosons, and CFT Data

The paper "Semiclassics, Goldstone Bosons, and CFT Data" explores the semi-classical analysis of operators with large conserved charges in Conformal Field Theories (CFTs). The primary focus is on understanding how symmetry breaking patterns and the resulting Goldstone modes can reveal universal aspects of CFTs in these regimes, extending previous insights gained in specific examples. This research forms part of a broader effort to leverage symmetry principles in unveiling the structure of CFTs, which are cornerstones in theoretical physics and widely applicable in describing critical phenomena in condensed matter and high-energy physics.

Overview

The core idea under investigation is the connection between operators with large global charges in a CFT and semiclassical states on the cylinder. This approach is rooted in the radial quantization mapping, where the problem of determining the spectrum of a CFT on flat space is related to studying the same theory compactified on a cylinder. In this compactified picture, states correspond to operators and their energies to operator dimensions. Large charge operators correspond to states with sizable charge densities, which can be analyzed semiclassically because the charge serves as a control parameter enabling perturbative expansions.

Main Contributions

1. Semiclassical Approach: The authors elaborate on the semiclassical treatment initially proposed by Hellerman et al., wherein the properties of operators with large internal U(1) charges can be examined using effective field theory techniques. By considering states with a large charge density, the problem is effectively reduced to studying superfluid phases on the cylinder.
2. Goldstone Bosons and Effective Theories: The research further clarifies how Goldstone bosons arise due to spontaneous symmetry breaking in these semiclassics. The coset construction method is used to systematically develop effective Lagrangians for the Goldstone modes, which capture the low-energy dynamics and allow for controlled expansions around large charges.
3. Extension to Non-Abelian Symmetries: Beyond the abelian case, the authors extend considerations to non-Abelian symmetry groups, such as $U(1) \times U(1)$ and $SO(3)$. This extension is non-trivial, as it involves handling multiple Goldstone bosons and considering gaps between energy states that may not be universal or straightforward to compute.
4. Implications and Applications: By exploring various internal symmetry groups, the paper underscores the potential universality in the behavior of CFTs with large global symmetries. The scaling dimensions of operators, characteristically $\Delta_Q \sim Q^{3/2}$ for large charge $Q$, are determined up to coefficients that depend on specific CFT data, which are non-perturbative and require special assumptions or additional methods to estimate or compute.
5. Three and Four-Point Functions: The paper also explores $n$-point correlation functions with two large charge operator insertions, showcasing how semiclassical techniques can predict these interactions. These predictions are compared against OPE expansions, offering insights into the structure constants and scaling laws that typify CFT correlations.

Speculative Outlook

For the future, more robust analytic and numeric methods could potentially extend these results or apply them more widely within CFT contexts. The insights gained from this semiclassical approach can provide informative benchmarks for testing numerical lattice equivalents of CFTs, as well as applications in condensed matter systems described by non-relativistic analogs of these theories.

Furthermore, understanding the interplay between large charge expansions and seemingly unrelated aspects of CFTs, like modular invariance or higher-genus surface considerations, might illuminate deeper connections within the field and across different formulations and theories.

This paper's approach to combining symmetry considerations with effective field theories holds promise for advancing the understanding of CFT structures operating in large quantum number regimes and offers a versatile framework for extensions potentially crossing into other domains in theoretical physics.