[Paper Review] Bootstrapping Boundary QED Part I

Samuel Bartlett-Tisdall, Christopher P. Herzog+1 more|arXiv (Cornell University)|Dec 12, 2023

Black Holes and Theoretical Physics4 citations

TL;DR

This paper uses the numerical conformal bootstrap to study boundary quantum electrodynamics (QED), a 4D photon coupled to 2+1D conformal matter on a boundary. By leveraging Maxwell equations and locality constraints, it derives exact relations for boundary three-point functions, proves a spin-two gap bound of less than 1.05 in 3D CFTs with U(1) symmetry, and derives an upper bound on the displacement operator two-point function, constraining the boundary anomaly and coupling strength.

ABSTRACT

We use the numerical conformal bootstrap to study boundary quantum electrodynamics, the theory of a four dimensional photon in a half space coupled to charged conformal matter on the boundary. This system is believed to be a boundary conformal field theory with an exactly marginal coupling corresponding to the strength of the interaction between the photon and the matter degrees of freedom. In part one of this project, we present three results. We show how the Maxwell equations put severe constraints on boundary three-point functions involving two currents and a symmetric traceless tensor. We use semi-definite programming to show that any three dimensional conformal field theory with a global U(1) symmetry must have a spin two gap less than about 1.05. Finally, combining a numerical bound on an OPE coefficient and some Ward identities involving the current and the displacement operator, we bound the displacement operator two-point function above. This upper bound also constrains a boundary contribution to the anomaly in the trace of the stress tensor for these types of theories.

Motivation & Objective

• To investigate boundary quantum electrodynamics (QED) as a boundary conformal field theory (bCFT) with an exactly marginal coupling.
• To apply the numerical conformal bootstrap to correlation functions involving boundary currents and the displacement operator in boundary QED.
• To derive constraints on the boundary operator spectrum using bulk equations of motion and locality.
• To bound the displacement operator two-point function and relate it to the anomaly in the stress tensor trace.
• To establish a connection between the optical response of boundary QED and the optical conductivity of materials like graphene.

Proposed method

• Using the numerical conformal bootstrap with semi-definite programming to analyze correlation functions in boundary QED.
• Applying the Maxwell equations to constrain boundary three-point functions involving two currents and a symmetric traceless tensor.
• Deriving exact relations between boundary current-current-scalar three-point functions via regularity constraints on bulk-boundary-boundary correlators.
• Implementing Ward identities and conformal invariance to fix the form of current-current-displacement three-point functions.
• Using the bulk OPE of the Maxwell field with itself to relate the displacement operator to the boundary limit of bulk operators like $F^2$, $F\widetilde{F}$, and $T_{\mu\nu}$.
• Relating the OPE coefficients of the displacement operator to the anomaly coefficient and coupling strength via Ward identities and boundary limits.

Experimental results

Research questions

• RQ1 What constraints do the Maxwell equations impose on boundary three-point functions involving two currents and a symmetric traceless tensor?
• RQ2 What is the upper bound on the displacement operator two-point function in boundary QED, and how does it constrain the boundary anomaly?
• RQ3 Can the numerical conformal bootstrap be applied to boundary QED despite positivity issues in bulk correlation functions?
• RQ4 How does the spin-two gap in 3D CFTs with U(1) symmetry relate to the coupling strength in boundary QED?
• RQ5 To what extent can boundary QED serve as a model for the optical conductivity of graphene?

Key findings

• The Maxwell equations imply the existence of exactly two boundary currents—electric and magnetic—arising from the operator expansion of the bulk Maxwell field.
• A semi-definite programming analysis shows that any 3D CFT with a global U(1) symmetry must have a spin-two gap less than approximately 1.05.
• An upper bound is derived for the displacement operator two-point function, which constrains the boundary contribution to the trace anomaly of the stress tensor.
• The coefficient of the current-current-displacement three-point function is related to the coupling strength and the anomaly coefficient via $\gamma_{EE}^{1} = \frac{2}{\pi^2}\tau_{EE} + \frac{\kappa}{\pi^2}\left(\frac{C_D}{C_D^{\text{free}}}-1\right)$.
• The boundary limit of bulk correlators $\langle F_{\mu\nu}F_{\rho\sigma}D\rangle$ is used to fix the form of the boundary three-point functions and relate them to physical observables.
• The analysis provides a framework to connect boundary QED to the optical conductivity of graphene, potentially resolving long-standing puzzles in its response functions.