Stress tensor and current correlators of interacting conformal field theories in 2+1 dimensions: Fermionic Dirac matter coupled to U(1) gauge field

TL;DR

This work computes two central universal quantities, the central charge $C_T$ and the flavor current normalization $C_J$, for $N_F$ Dirac fermions coupled to a $U(1)$ gauge field in 2+1 dimensions up to next-to-leading order in $1/N_F$, i.e., $C_T/N_F=\frac{1}{256}[1+\frac{1}{N_F}(\tilde{C}_T^{(1)}+\frac{104}{15\pi^2})]$ and $C_J=\frac{1}{16}[1+\frac{1}{N_F}(\tilde{C}_J^{(1)}-\frac{40}{9\pi^2})]$, with numerical coefficients $\approx0.2870$ and $0.1429$, respectively. The authors implement a momentum-space, tensor-integral approach (Tensoria) to evaluate the relevant Feynman diagrams, verify the cancellation of longitudinal and logarithmic divergences consistent with current and stress-tensor conservation, and provide explicit analytic and numerical expressions for the diagrammatic contributions. The positive $1/N_F$ corrections contrast with some bosonic cases and carry implications for response functions, entanglement measures, and bootstrap/ddualities in 3D CFTs, offering precise benchmarks for numerical simulations and theoretical cross-checks in strongly correlated systems.

Abstract

We compute the central charge $C_T$ and universal conductivity $C_J$ of $N_F$ fermions coupled to a $U(1)$ gauge field up to next-to-leading order in the $1/N_F$ expansion. We discuss implications of these precision computations as a diagnostic for response and entanglement properties of interacting conformal field theories for strongly correlated condensed matter phases and conformal quantum electrodynamics in $2+1$ dimensions.

Figures (6)

• Figure 1: Feyman rules for $N_F$ Dirac fermions coupled to $U(1)$ gauge field in Eq. (\ref{['eq:bare_action']}).
• Figure 2: Feyman rule for the current vertex. $T^{\ell}$ is a generator of the SU$(N_F)$.
• Figure 3: Feyman diagrams contributing to the current current correlator to order $1/N_F$. Diagram (0) is the leading order contribution and the only one that survives the $N_F \rightarrow \infty$ limit. Diagram (1) is the vertex correction, diagram (2) the self-energy correction that comes with a factor of $a_2 = 2$.
• Figure 4: Feyman rules for the stress tensor vertices.
• Figure 5: Feynman diagrams contributing to the stress energy tensor correlator to next-to-leading order in $1/N_F$. Only diagram (0) survives in the $N_F\rightarrow \infty$ limit. Diagrams (2) and (4) come with a factor of $a_2=2$, $a_4=2$, respectively. The factors for the other graphs are unity $a_i=1$. The numerical values and logarithmic singularities for each of these graphs are exhibited in Table \ref{['tab:c_t']}.